Quiz: Statistics for Management — 64 questions

1. Which statement best distinguishes statistics as a discipline from statistics as numerical results?

A. Statistics as a discipline records observations, whereas statistics as results identify the population being studied.

B. Statistics as a discipline displays categories, whereas statistics as results determine which variables should be measured.

C. Statistics as a discipline provides methods for analyzing data, whereas statistics as results are calculated numerical summaries.

D. Statistics as a discipline lists measured values, whereas statistics as results explain how samples were selected.

Answer: (C)

Explanation

Statistics as a discipline includes reasoning, tools, and methods for analyzing data, while statistics in the plural are numerical results produced from calculations. Treating the discipline as a record of observations confuses analytical methods with the data being analyzed.

2. Why does an isolated value provide limited statistical meaning?

A. Its meaning depends on placing the value in a chart with a numerical horizontal axis.

B. Its meaning depends on converting the value into a percentage before reporting it.

C. Its meaning depends on proving that the value came from a randomly selected sample.

D. Its meaning depends on the context describing what was measured and how the value varies.

Answer: (D)

Explanation

Data are values that vary in a context, so the measured subject, variable, and circumstances are needed for interpretation. A percentage, chart, or random sample may be useful in some analyses but does not by itself define what an isolated value represents.

3. A researcher plans an analysis, creates graphs, and explains the findings to a public audience; which aspect of doing statistics is illustrated by this sequence?

A. Collecting numerical observations, removing contextual information, and reporting calculations without interpretation

B. Thinking before analysis, communicating results mathematically and visually, and explaining them clearly

C. Selecting a population, assigning identifiers, and arranging categories in decreasing frequency

D. Measuring variables, replacing sample values with forecasts, and presenting conclusions without displays

Answer: (B)

Explanation

Doing statistics includes planning and thinking before analysis, using mathematics and charts to show results, and communicating interpretations so others understand them correctly. The other sequences omit or contradict the required role of context, displays, or explanation.

4. Which set of questions provides the full context needed to understand a dataset?

A. Which chart was selected, which formula was used, which software ran it, and which color marked the results

B. Who was studied, what was measured, why and how data were collected, and when and where collection occurred

C. How many variables were included, how many tables were printed, which average was largest, and which sample was shortest

D. Whether the values were rounded, whether the categories were alphabetized, whether the graph had a title, and whether results were published

Answer: (B)

Explanation

The six contextual questions concern who, what, why, how, when, and where. Questions about software, graph colors, or formatting may concern presentation but do not establish the dataset’s substantive context.

5. A school records students’ favorite sport and weekly hours of exercise; how should these two variables be classified?

A. Both variables are quantitative because each is recorded for individual students.

B. Favorite sport is qualitative, while weekly exercise hours are quantitative.

C. Favorite sport is quantitative, while weekly exercise hours are qualitative.

D. Both variables are qualitative because they describe student characteristics.

Answer: (B)

Explanation

Favorite sport identifies categories, making it qualitative, whereas weekly exercise hours measure an amount with numerical meaning, making it quantitative. Recording a variable with numbers does not make a category variable quantitative.

6. Why is a student ID number treated as categorical rather than quantitative?

A. The ID values describe how much time or distance each student has accumulated.

B. Each ID must be converted into a percentage before it can be analyzed statistically.

C. The ID values are written with digits but cannot be stored in a numerical column.

D. Arithmetic operations on the ID values do not represent meaningful measured amounts.

Answer: (D)

Explanation

A numerical variable is quantitative only when calculations with its values make sense as measurements, and arithmetic on student IDs does not have that interpretation. The presence of digits does not turn an identifier into a measured amount.

7. What condition defines a simple random sample of size $n$?

A. The population is divided into groups and one entire group is selected for study.

B. Every different sample of $n$ experimental units has an equal chance of being selected.

C. Units are selected because they are easiest for the researcher to contact.

D. Each experimental unit is selected at a fixed interval after the first unit is chosen.

Answer: (B)

Explanation

A simple random sample gives every possible sample of size $n$ the same selection probability. Fixed intervals describe systematic sampling, easy access describes convenience sampling, and selecting whole groups describes cluster sampling.

8. Which display is most appropriate for showing the distribution of a quantitative variable such as commute time?

A. A pie chart with slices representing each individual commute time

B. A histogram with adjacent bars representing equal-width intervals of commute time

C. A bar chart with separated bars representing numerical intervals as categories

D. A frequency table that replaces each commute time with a qualitative label

Answer: (B)

Explanation

Quantitative values such as commute times are well represented by a histogram, which groups values into intervals. A pie chart is suited to categories, and separated bars are characteristic of categorical displays rather than quantitative intervals.

9. How does a histogram represent the distribution of a quantitative variable?

A. It displays individual observations as labels and uses slices to show their relative contributions.

B. It lists categories alphabetically and uses separated bars to show the percentage in each category.

C. It orders categories from largest to smallest and places the tallest bar at the left.

D. It divides values into equal-width intervals and uses each bar’s height for frequency or relative frequency.

Answer: (D)

Explanation

A histogram partitions quantitative values into equal-width class intervals, with bar heights showing frequency or relative frequency. Alphabetical categories, decreasing category order, and slices describe other displays rather than the defining structure of a histogram.

10. Which calculation defines the sample mean for observations $x_1, x_2, \ldots, x_n$?

A. The sum of all observations divided by the number of observations

B. The difference between the third and first quartiles

C. The middle observation after the values are placed in order

D. The largest observation minus the smallest observation

Answer: (A)

Explanation

The sample mean is computed by summing every observation and dividing by $n$. The middle observation describes the median, not the mean.

11. For the ordered data set $2, 3, 4, 5, 20$, which measure of center is generally more resistant to the extreme value $20$?

A. The full range

B. The standard deviation

C. The median

D. The sample mean

Answer: (C)

Explanation

The median is resistant to extreme observations, so it remains closer to the center of the smaller values. The mean is sensitive to the unusually large value and is pulled upward.

12. Two data sets have the same mean, but the second has a larger standard deviation; what does this indicate about the second data set?

A. Its middle observation is closer to the mean

B. Its observations are more spread out around the mean

C. Its observations have a smaller minimum value

D. Its third quartile equals its first quartile

Answer: (B)

Explanation

Standard deviation measures variation around the mean, so a larger value indicates greater dispersion. It does not directly specify the minimum, median, or quartiles.

13. A distribution contains outliers and several clusters; which summary is generally preferred for describing its center and spread?

A. The median together with the interquartile range

B. The range together with the sample mean

C. The mean together with the standard deviation

D. The standard deviation together with the full range

Answer: (A)

Explanation

For distributions with outliers or clusters, the median and IQR are generally preferred because they are less sensitive to extreme values. The mean and standard deviation can be distorted by such features.

14. What does a contingency table organize?

A. Ranks for observations within one ordered sample

B. Means for pairs of quantitative variables

C. Predictions from a single numerical variable

D. Counts for combinations of two categorical variables

Answer: (D)

Explanation

A contingency table places one categorical variable in rows and another in columns, displaying counts for their combinations. Means and predictions belong to quantitative-data analyses rather than this table structure.

15. If the conditional percentages for one categorical variable are similar across every category of the other variable, what conclusion is supported?

A. The row totals must equal the column totals

B. The two categorical variables have a perfect association

C. The two categorical variables are likely independent

D. The sample contains a hidden numerical variable

Answer: (C)

Explanation

Independence is indicated when the distribution of one variable is similar across categories of the other. Perfect association would require a much stronger pattern, while equal margins are not required.

16. A contingency table has a row total of $40$, a column total of $30$, and a sample size of $200$. Under independence, what is the expected count for that cell?

A. $6$

B. $15$

C. $70$

D. $20$

Answer: (A)

Explanation

Under independence, the expected count is $E_{ij}=\frac{R_iC_j}{n}=\frac{40\times30}{200}=6$. The other values do not result from multiplying the relevant margins and dividing by the sample size.

17. What does Cramer’s V measure in an analysis of two categorical variables?

A. The difference between two marginal percentages

B. The standardized strength of their association

C. The expected count under independence

D. The number of observations in the largest cell

Answer: (B)

Explanation

Cramer’s V is a standardized association coefficient based on chi-squared, ranging from 0 for no association to 1 for perfect association. It is not a cell count or a direct difference between percentages.

18. In a scatterplot of study time and exam score, which assignment matches the roles of the variables in a predictive analysis?

A. Both variables are assigned the role of $Y$

B. Study time is $X$ and exam score is $Y$

C. Exam score is $X$ and study time is $Y$

D. Both variables are assigned the role of $X$

Answer: (B)

Explanation

The explanatory variable is placed on the $X$ axis, while the response variable is represented by $Y$. In this setting, study time is used to explain or predict exam score.

19. What does a correlation coefficient of $r=-0.85$ indicate?

A. A weak positive curved association

B. A perfect causal relationship

C. A strong positive linear association

D. A strong negative linear association

Answer: (D)

Explanation

A value of $r=-0.85$ is close to $-1$, indicating a strong negative linear association. Correlation describes direction and strength of linear association, not causation.

20. Why does observing a strong correlation between two variables fail to establish that one causes the other?

A. The association may reflect coincidence or a lurking variable

B. A scatterplot contains only categorical observations

C. Correlation coefficients cannot describe positive or negative direction

D. Linear models cannot be used with quantitative variables

Answer: (A)

Explanation

Correlation can arise from coincidence, direct causality, or a common underlying cause called a lurking variable. The other claims contradict the role and interpretation of correlation.

21. A regression model was fitted using values of $x$ between $10$ and $50$. Predicting the response at $x=30$ is called what?

A. Residual calculation from the response value

B. Extrapolation beyond the model’s observed domain

C. Causal identification from the fitted line

D. Interpolation within the model’s observed domain

Answer: (D)

Explanation

Predicting at $x=30$ is interpolation because $30$ lies within the range used to construct the model. Extrapolation would involve predicting below $10$ or above $50$ and requires caution.

22. What quantity does the coefficient of determination $r^2$ represent?

A. The percentage of observations predicted with exact accuracy

B. The percentage of total variation accounted for by the model

C. The strength and direction of association between two variables

D. The average vertical distance from observations to the fitted line

Answer: (B)

Explanation

The coefficient of determination compares variation explained by the model with total variation, so it expresses the proportion accounted for. The correlation coefficient $r$ measures association, while the standard error measures typical vertical deviation.

23. In the manatee-deaths and powerboat-registrations model, what does $r^2=0.895$ imply?

A. The model predicts 89.5% of individual death counts exactly

B. The model accounts for 89.5% of the variation in manatee deaths

C. The correlation between the variables equals 0.895 deaths

D. The model leaves 89.5% of the variation unexplained

Answer: (B)

Explanation

An $r^2$ value of 0.895 means that 89.5% of the variation in manatee deaths is accounted for by the model. It does not mean that 89.5% of individual observations are predicted exactly.

24. Which residual-plot pattern provides the strongest evidence that a fitted regression model is appropriate?

A. Roughly constant horizontal scatter with no bend and few outliers

B. A widening spread that increases steadily from left to right

C. A tight cluster containing many unusually distant points

D. A curved pattern showing residuals rise and then fall

Answer: (A)

Explanation

A useful residual plot has roughly constant scatter, no systematic curvature, and few or no outliers. A widening spread, bend, or many distant points signals a potential problem with the model.

25. What does the standard error of regression measure?

A. The direction in which the association between variables changes

B. How much the response values vary around the fitted line

C. The fraction of response variation explained by the regression model

D. How much the explanatory variable varies around its sample mean

Answer: (B)

Explanation

The standard error of regression measures the typical variation of response values around the fitted line and is also called the root mean squared error. The fraction explained is measured by $r^2$, not by the standard error.

26. What does the law of large numbers describe when a random trial is repeated many times?

A. Each individual trial becomes more likely to produce the long-run result

B. The event's observed proportion approaches a long-run empirical probability

C. The theoretical probability changes as additional trials are observed

D. Short sequences tend to balance outcomes with nearly equal frequencies

Answer: (B)

Explanation

The law of large numbers says that the observed proportion of an event approaches a single long-run empirical probability as trials accumulate. It concerns long-run behavior rather than guaranteeing balance in a short sequence.

27. A study observes event $A$ in 18 of 60 trials. What is its empirical probability?

A. $3.33$

B. $0.30$

C. $0.60$

D. $0.42$

Answer: (B)

Explanation

Empirical probability is the number of observed occurrences divided by the number of trials, so $P(A)=\frac{18}{60}=0.30$. A theoretical probability would instead come from a probability model rather than these observed counts.

28. If $P(A)=0.37$, what is the probability of the complement of $A$?

A. $-0.37$

B. $0.37$

C. $0.63$

D. $1.37$

Answer: (C)

Explanation

The complement rule gives $P(A^C)=1-P(A)$, so $P(A^C)=1-0.37=0.63$. The value $0.37$ is the probability of $A$ itself, not its complement.

29. Events $A$ and $B$ are disjoint with $P(A)=0.28$ and $P(B)=0.41$. What is $P(A\text{ or }B)$?

A. $0.28$

B. $0.69$

C. $0.1148$

D. $0.41$

Answer: (B)

Explanation

For disjoint events, the addition rule is $P(A\text{ or }B)=P(A)+P(B)$, giving $0.28+0.41=0.69$. Multiplication would address a different probability question and is not the disjoint-event addition rule.

30. What does a random variable assign to the outcomes of a random event?

A. Numerical values

B. Sets of possible outcomes

C. Fixed probabilities for every experiment

D. Descriptions without numerical meaning

Answer: (A)

Explanation

A random variable assigns numerical values to outcomes of a random event. An event is a set of outcomes, so it is not itself the numerical assignment made by a random variable.

31. Which variable is discrete rather than continuous?

A. The exact temperature of a chemical sample

B. The time required to complete a delivery

C. The number of defective items in a batch

D. The measured height of a randomly selected plant

Answer: (C)

Explanation

The number of defective items is a countable quantity, making it a discrete random variable. Temperature, delivery time, and measured height can take values throughout numerical intervals and are continuous variables.

32. Which condition must a probability model for a discrete random variable satisfy?

A. Each probability is positive and their sum can exceed 1

B. Every possible value has the same probability and the mean is 1

C. Every probability is nonnegative and all probabilities sum to 1

D. The probabilities may be negative when outcomes are rare

Answer: (C)

Explanation

A valid discrete probability model requires $p(x)\ge0$ for every value and $\sum p(x)=1$. Equal probabilities are not required, and probabilities cannot be negative or sum to more than 1.

33. A discrete random variable has values $1,2,4$ with probabilities $0.2,0.5,0.3$, respectively. What is its expected value?

A. $5.0$

B. $2.0$

C. $2.7$

D. $2.1$

Answer: (D)

Explanation

The expected value is the probability-weighted sum, $E(X)=1(0.2)+2(0.5)+4(0.3)=2.1$. It is not obtained by simply averaging the listed values without considering their probabilities.

34. Which condition defines a sequence of Bernoulli trials?

A. Each trial has success or failure, with outcomes determined by earlier trials

B. Each trial has several outcomes, with probabilities changing over time

C. Each trial has success or failure, with fixed probability and independence

D. Each trial has continuous outcomes, with a probability density for results

Answer: (C)

Explanation

Bernoulli trials require two possible outcomes, a fixed success probability, and independence between successive trials. A general random experiment may have more outcomes or changing and dependent probabilities.

35. A binomial experiment has $n=100$ trials and success probability $p=0.30$. What are its expected value and standard deviation?

A. $\mu=30$ and $\sigma=\sqrt{30}\approx5.48$

B. $\mu=70$ and $\sigma=\sqrt{30}\approx5.48$

C. $\mu=0.30$ and $\sigma=\sqrt{70}\approx8.37$

D. $\mu=30$ and $\sigma=\sqrt{21}\approx4.58$

Answer: (D)

Explanation

For a binomial variable, $\mu=np=100(0.30)=30$ and $\sigma=\sqrt{npq}=\sqrt{100(0.30)(0.70)}=\sqrt{21}$. Using the failure probability as the mean would confuse successes with failures.

36. Which situation is most naturally modeled by a Poisson distribution?

A. The proportion of defective items in a shipment

B. The number of printing errors in a fixed-length document

C. The measured weight of randomly selected packages

D. The number of correct answers on a fixed quiz

Answer: (B)

Explanation

A Poisson distribution models the number of rare events occurring within a specified interval of time, length, area, or a similar measurement. Correct-answer counts are typically binomial, proportions are not event counts, and package weights are continuous measurements.

37. For a continuous random variable, what does the area under a density curve between two values represent?

A. The expected value of every observation in that interval

B. The probability that the variable equals the interval midpoint

C. The standard deviation calculated from values in that interval

D. The probability that the variable falls within that interval

Answer: (D)

Explanation

For a continuous variable, probability is represented by the area under the density curve across an interval. The probability of one exact value is effectively zero, so the midpoint does not represent the interval's probability.

38. Which description best characterizes a Normal model written as $N(\mu,\sigma)$?

A. A continuous, approximately unimodal and symmetric distribution

B. A discrete, strongly right-skewed distribution with a fixed mode

C. A continuous, multimodal distribution with unequal tails

D. A discrete, uniform distribution in which all values are equally likely

Answer: (A)

Explanation

A Normal model is continuous, approximately unimodal, and symmetric, with mean $\mu$ and standard deviation $\sigma$. A strongly skewed or multimodal distribution is not appropriately described by this model.

39. In a Normal model, approximately what percentage of observations lie within two standard deviations of the mean?

A. About 68%

B. About 90%

C. About 95%

D. About 99.7%

Answer: (C)

Explanation

The 68-95-99.7 rule places approximately 95% of Normal-model observations within two standard deviations of the mean. About 68% corresponds to one standard deviation, while 99.7% corresponds to three.

40. A measurement is $x=74$ from a population with mean $\mu=68$ and standard deviation $\sigma=3$. What is its z-score?

A. $z=2$

B. $z=\frac{68}{3}$

C. $z=-2$

D. $z=\frac{74}{3}$

Answer: (A)

Explanation

Standardizing gives $z=\frac{x-\mu}{\sigma}=\frac{74-68}{3}=2$, so the measurement is two standard deviations above the mean. A negative score would indicate a value below the mean.

41. Which statement correctly distinguishes a population parameter from a sample statistic?

A. A parameter describes a sampling distribution, whereas a statistic describes an interval

B. A parameter describes a population, whereas a statistic is calculated from a sample

C. A parameter is calculated from a sample, whereas a statistic describes the population

D. A parameter measures variability, whereas a statistic measures central tendency

Answer: (B)

Explanation

A parameter is a numerical description of an entire population, while a statistic is calculated from observations in a sample. Either type of quantity can describe central tendency or variability, so that distinction is not based on what it measures.

42. What does the sampling distribution of a statistic describe?

A. The probability distribution of population parameters across different populations

B. The values of one statistic calculated repeatedly from one unchanged observation

C. The distribution of individual observations across the entire population

D. The probability distribution of that statistic across all samples of a given size

Answer: (D)

Explanation

A sampling distribution shows how a statistic varies across all possible samples of a specified size. It differs from the population distribution, which describes individual observations rather than statistics from samples.

43. What does the Central Limit Theorem imply about sample means as sample size increases?

A. Their sampling distribution becomes approximately Normal

B. Their individual observations become identical to the population mean

C. Their sampling distribution becomes more strongly skewed

D. Their population distribution becomes exactly uniform

Answer: (A)

Explanation

The Central Limit Theorem states that the sampling distribution of a random sample mean becomes approximately Normal as sample size increases, often with $n\ge30$ depending on the original distribution. It concerns the distribution of sample means, not the transformation of individual observations.

44. For a quantitative variable with population standard deviation $\sigma$, how does the standard error of the sample mean change when the sample size is increased from $n$ to $4n$?

A. It becomes half as large

B. It becomes four times as large

C. It remains unchanged

D. It becomes twice as large

Answer: (A)

Explanation

The standard error is $\sigma/\sqrt{n}$, so replacing $n$ with $4n$ changes it to $\sigma/(2\sqrt{n})$. The population standard deviation describes individual observations and does not change through this sampling adjustment.

45. A population distribution is strongly skewed rather than normal. Which condition most directly supports using a normal model for the distribution of its sample mean?

A. Using a sufficiently large random sample

B. Using a sample whose observations are all identical

C. Selecting a sample with a population mean of zero

D. Measuring a variable with a small standard deviation

Answer: (A)

Explanation

For a non-normal population, the sampling distribution of the mean tends toward normality as the sample size increases. A small standard deviation or a particular mean does not determine the shape of that sampling distribution.

46. According to the Central Limit Theorem, which sample size would generally provide an adequate normal approximation for a sample mean when the population is not normal?

A. A sample size of at least 30

B. A sample size equal to the population size

C. A sample size of 5 or fewer

D. A sample size of exactly 2

Answer: (A)

Explanation

The Central Limit Theorem says that the normal approximation improves as $n$ increases and is usually adequate when $n\ge30$. Very small samples may retain important features of the original non-normal population.

47. What does the sampling distribution of a proportion describe?

A. The standard deviation of one particular sample proportion

B. The proportions from all possible samples of a fixed size

C. The observations recorded from one selected individual

D. The population proportion measured in one complete population

Answer: (B)

Explanation

A sampling distribution of a proportion consists of the proportions obtained from all possible samples of size $n$. A proportion calculated from one sample is a sample statistic, not the entire sampling distribution.

48. If a population has proportion $p=0.40$ and samples have size $n=100$, what is the standard error of the sample proportion?

A. $0.060$

B. $0.049$

C. $0.240$

D. $0.400$

Answer: (B)

Explanation

Using $SD(\hat{p})=\sqrt{p(1-p)/n}$ gives $\sqrt{(0.40)(0.60)/100}\approx0.049$. The value $0.400$ is the population proportion, not the sampling standard error.

49. Which condition supports using an approximately normal sampling distribution for a sample proportion?

A. The sample size is smaller than both expected counts

B. Both $np\ge15$ and $n(1-p)\ge15$

C. The population standard deviation equals one

D. The sample mean equals the population mean

Answer: (B)

Explanation

The normal approximation for a sample proportion requires at least 15 expected observations in each category, expressed as $np\ge15$ and $n(1-p)\ge15$. The mean and standard deviation conditions concern other sampling distributions or do not establish normality here.

50. A researcher wants to estimate the average income of all residents in a city. What is the target parameter?

A. The standard error of the sample mean

B. The average income in one selected sample

C. The number of residents surveyed

D. The unknown population mean income

Answer: (D)

Explanation

The target parameter is the unknown population quantity being estimated, which here is the citywide mean income. The average from a sample is an estimator calculated from observed data rather than the population parameter itself.

51. Which description best distinguishes a point estimator from an interval estimator?

A. A point estimator uses a population value, whereas an interval estimator uses no data

B. A point estimator gives one value, whereas an interval estimator gives a range

C. A point estimator applies to proportions, whereas an interval estimator applies to means

D. A point estimator measures variation, whereas an interval estimator gives one exact value

Answer: (B)

Explanation

A point estimator uses sample data to produce a single numerical estimate, while an interval estimator produces a range that accounts for sampling variation. Both types can be used for parameters such as means or proportions.

52. Why does a confidence interval provide a range rather than a single estimate of a population parameter?

A. It guarantees that every observed value equals the parameter

B. It replaces the target parameter with a known population statistic

C. It accounts for variation among possible random samples

D. It describes the distribution of individual observations

Answer: (C)

Explanation

An interval estimator uses sample data to give a range that reflects sample-to-sample variation. It does not make the population parameter known or describe the distribution of individual observations.

53. A confidence interval is constructed using a point estimate of 18, a critical value of 2, and a standard error of 3. What is the resulting interval?

A. $6\text{ to }30$

B. $15\text{ to }21$

C. $16\text{ to }20$

D. $12\text{ to }24$

Answer: (D)

Explanation

The general formula is $\text{Point Estimate}\pm(\text{Critical Value})(\text{Standard Error})$, so the interval is $18\pm(2)(3)=18\pm6$, or 12 to 24. The narrower alternatives do not apply the full critical-value times standard-error margin.

54. When the population standard deviation is known, which expression gives a confidence interval for a population mean?

A. $\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$

B. $\bar{x}\pm t_{\alpha/2}\frac{s}{\sqrt{n}}$

C. $\hat{p}\pm z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

D. $\bar{x}\pm z_{\alpha}\frac{s}{n}$

Answer: (A)

Explanation

The z-interval uses the sample mean, the known population standard deviation, and the standard normal critical value. The t-based expression is used when the population standard deviation is unknown and is replaced by the sample standard deviation.

55. Which set of conditions best justifies a z-interval for a population mean?

A. A random sample, known population standard deviation, and a sufficiently large sample

B. A convenience sample, known population standard deviation, and a large sample

C. A random sample, estimated population standard deviation, and any sample size

D. A representative sample, unknown population standard deviation, and a small sample

Answer: (A)

Explanation

A z-interval for a mean requires random sampling, known population standard deviation, and a large sample such as $n\ge30$ to support approximate normality. Knowing $\sigma$ does not compensate for a nonrandom sample or an inadequate sample size.

56. A population standard deviation is unknown, and a sample of size $n$ is used to estimate the population mean. Which interval is appropriate?

A. $\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$ with $df=n$

B. $\bar{x}\pm z_{\alpha/2}\frac{s}{n}$ with $df=n-1$

C. $\bar{x}\pm t_{\alpha/2}\frac{s}{\sqrt{n}}$ with $df=n-1$

D. $\hat{p}\pm t_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ with $df=n-1$

Answer: (C)

Explanation

When $\sigma$ is unknown, the mean interval substitutes $s$ and uses a Student t critical value with $df=n-1$. The z-based expression assumes that the population standard deviation is known.

57. Which expression gives a confidence interval for a population proportion?

A. $\hat{p}\pm z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

B. $\bar{x}\pm z_{\alpha/2}\sqrt{\frac{\bar{x}(1-\bar{x})}{n}}$

C. $\hat{p}\pm t_{\alpha/2}\frac{s}{\sqrt{n}}$

D. $\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$

Answer: (A)

Explanation

A proportion interval uses the sample proportion $\hat{p}$ in both the point estimate and the estimated standard error. The mean interval instead uses the sample mean and a standard deviation term.

58. For a confidence interval for a population proportion, which conditions should be checked for the observed sample?

A. A convenience sample with $n\hat{p}\ge10$ and $n(1-\hat{p})\ge10$

B. A representative sample with $n\bar{x}\ge15$ and $n(1-\bar{x})\ge15$

C. A random sample with $n p_0\ge15$ and $n(1-p_0)\ge15$

D. A random or representative sample with $n\hat{p}\ge15$ and $n(1-\hat{p})\ge15$

Answer: (D)

Explanation

A proportion confidence interval requires a random or representative sample and sufficiently large expected success and failure counts based on $\hat{p}$. The null value $p_0$ is used for checking counts in a hypothesis test, not for constructing this interval.

59. What does the margin of error represent in a confidence interval?

A. The amount added to and subtracted from the point estimate

B. The population standard deviation used in a mean calculation

C. The difference between the sample size and the confidence level

D. The probability that the population parameter equals the point estimate

Answer: (A)

Explanation

The margin of error measures estimation imprecision by extending the point estimate in both directions to create the interval. It is not a probability that the parameter equals the estimate.

60. What generally happens when the confidence level for an interval is increased?

A. The interval becomes narrower, increasing certainty while improving precision

B. The point estimate changes, while the interval width remains fixed

C. The interval becomes wider, increasing certainty while reducing precision

D. The sample proportion changes, while the confidence level remains fixed

Answer: (C)

Explanation

A higher confidence level requires a larger critical value, producing a wider interval. This gives greater certainty about capturing the parameter but reduces precision because the estimate covers a broader range.

61. Which statement correctly describes the null hypothesis in a one-sample hypothesis test?

A. It is a claim about the observed sample statistic, often expressing the direction of sampling error

B. It is a claim about a population parameter, often expressing a status quo or hypothesized equality

C. It is the probability of obtaining a test statistic at least as extreme as the observed value

D. It is the population-parameter claim supported after the null hypothesis is rejected

Answer: (B)

Explanation

The null hypothesis is a claim about a population parameter and commonly represents a status quo or equality to a hypothesized value. A sample statistic supplies evidence for evaluating that claim but is not itself the null hypothesis.

62. When is the alternative hypothesis supported in a one-sample test?

A. When the population parameter is replaced by the observed sample value

B. When the significance level is selected before stating the hypotheses

C. When the sample provides convincing evidence against the null hypothesis

D. When the sample statistic proves that the null hypothesis is true

Answer: (C)

Explanation

The alternative hypothesis states the population-parameter claim opposite to the null and is supported when the sample gives convincing evidence against the null. A test does not prove the null hypothesis true when evidence against it is insufficient.

63. What is the correct interpretation of failing to reject the null hypothesis?

A. There is insufficient evidence to support the alternative hypothesis

B. The sample provides strong evidence that the null hypothesis is incorrect

C. The alternative hypothesis has been rejected as false for the population

D. The null hypothesis has been proven true for the population

Answer: (A)

Explanation

Failing to reject the null means the evidence is insufficient to support the alternative at the chosen standard. It does not establish that the null hypothesis is true or prove that the alternative is false.

64. A test uses significance level $\alpha=0.05$ and produces a p-value of $0.03$. What decision is appropriate?

A. Reject $H_0$ because the p-value is less than $\alpha$

B. Reject $H_0$ because the p-value is greater than $\alpha$

C. Fail to reject $H_0$ because the p-value equals the significance level

D. Fail to reject $H_0$ because the p-value is less than $\alpha$

Answer: (A)

Explanation

The p-value is compared with the significance level under the assumption that $H_0$ is true; a p-value below $\alpha$ leads to rejection of $H_0$. The stated values satisfy $0.03<0.05$, so the result provides evidence against the null hypothesis.