Quiz: Sampling Distributions and Confidence Intervals — 25 questions

1. Which statement best defines a population parameter?

A. A numerical measure estimating sampling variability

B. A probability distribution describing repeated samples

C. A numerical measure describing an entire population

D. A numerical measure calculated from a selected sample

Answer: (C)

Explanation

A population parameter summarizes a characteristic of all observations in the population and is often unknown. A sample statistic instead describes observations from a sample, not the entire population.

2. A researcher calculates the average income of 200 surveyed households. What does this calculated average represent?

A. A population standard deviation for household incomes

B. A sample statistic based on the surveyed households

C. A population parameter based on every household

D. A sampling distribution across repeated surveys

Answer: (B)

Explanation

The average is computed from observations in the surveyed sample, so it is a sample statistic. It would be a population parameter only if every household in the population had been included.

3. Which pairing correctly matches population quantities with their sample estimates?

A. Population mean $p$ with sample proportion $\hat{p}$; population proportion $\mu$ with sample mean $\bar{x}$

B. Population mean $\bar{x}$ with sample mean $\mu$; population proportion $\hat{p}$ with sample proportion $p$

C. Population mean $\mu$ with sample mean $\bar{x}$; population proportion $p$ with sample proportion $\hat{p}$

D. Population mean $\sigma$ with sample mean $s$; population proportion $n$ with sample proportion $\hat{p}$

Answer: (C)

Explanation

The population mean is denoted by $\mu$ and estimated by $\bar{x}$, while the population proportion $p$ is estimated by $\hat{p}$. The other pairings interchange population and sample notation or use unrelated quantities.

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

A. The probability distribution of a statistic across all possible samples of fixed size

B. The distribution of individual observations across the entire population

C. The probability distribution of population parameters across different variables

D. The collection of measurements obtained from one selected sample

Answer: (A)

Explanation

A sampling distribution shows how a statistic varies across all possible samples of a fixed size. A population distribution instead describes variation among individual observations.

5. A population has mean $\mu$ and standard deviation $\sigma$, and sample means are computed from samples of size $n$. Which expressions give the mean and standard error of the sampling distribution?

A. $E(\bar{x})=\frac{\mu}{n}$ and $SD(\bar{x})=\sigma\sqrt{n}$

B. $E(\bar{x})=\sigma$ and $SD(\bar{x})=\frac{\mu}{\sqrt{n}}$

C. $E(\bar{x})=n\mu$ and $SD(\bar{x})=\frac{\sigma}{n}$

D. $E(\bar{x})=\mu$ and $SD(\bar{x})=\frac{\sigma}{\sqrt{n}}$

Answer: (D)

Explanation

The sampling distribution of the mean is centered at the population mean, and its standard error is the population standard deviation divided by $\sqrt{n}$. The other expressions incorrectly scale the center or variability with sample size.

6. If the original population variable is normally distributed, what can be concluded about the sampling distribution of the mean?

A. It is normally distributed for any sample size

B. It becomes approximately uniform for small samples

C. It follows the population distribution only when the sample mean equals the population mean

D. It is normally distributed only when the sample has at least 30 observations

Answer: (A)

Explanation

When the original variable is normal, the sampling distribution of the mean is normal for every sample size. The threshold of about 30 is relevant to using the Central Limit Theorem for non-normal populations, not to this normal-population case.

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

A. Their sampling distribution is increasingly well approximated by a normal model

B. The individual observations become normally distributed within each sample

C. The variation among individual observations disappears from the population

D. Their population mean changes toward zero as more observations are collected

Answer: (A)

Explanation

The Central Limit Theorem concerns the sampling distribution of sample means, whose normal approximation improves as $n$ increases. It does not require the individual observations themselves to become normally distributed.

8. For a population with standard deviation 10 cm, what is the standard deviation of the sampling distribution of sample means when samples contain 50 individuals?

A. Approximately 2.24 cm

B. Exactly 10 cm

C. Approximately 1.41 cm

D. Approximately 70.71 cm

Answer: (C)

Explanation

The standard deviation of sample means is the population standard deviation divided by the square root of the sample size, giving $10/\sqrt{50}\approx1.41$ cm. The value 10 cm describes individual heights, not the variability of means from samples of 50.

9. A population has mean height 172 cm and standard deviation 10 cm. For samples of 50 heights, how unusual is a sample mean of at least 184 cm?

A. It is fairly common because its z-score is about 1.20

B. It is approximately impossible because its z-score is 8.48

C. It is moderately unusual because its z-score is about 2.00

D. It is slightly unusual because its z-score is about 0.85

Answer: (B)

Explanation

The sampling distribution has standard deviation about 1.41 cm, so a mean of 184 cm has a z-score of about 8.48 and probability approximately 0. A z-score near 2 would indicate a much less extreme result than this sample mean.

10. What does the sampling distribution of $\hat{p}$ represent?

A. The probability distribution of sample proportions from all possible samples of a fixed size

B. The single population proportion that remains fixed across repeated samples

C. The distribution of individual observations recorded within one selected sample

D. The collection of population parameters estimated from several different studies

Answer: (A)

Explanation

The sampling distribution of $\hat{p}$ describes how sample proportions vary across all possible samples of size $n$. The population proportion is a fixed parameter, not a distribution of sample results.

11. Under which conditions is the sampling distribution of $\hat{p}$ approximately normal?

A. When the population proportion is exactly 0.50

B. When $np\ge5$ and $n(1-p)\ge5$

C. When the sample size is at least 15 regardless of $p$

D. When $np\ge15$ and $n(1-p)\ge15$

Answer: (D)

Explanation

The approximation is considered valid when both expected counts, $np$ and $n(1-p)$, are at least 15. A sample-size threshold without reference to the population proportion does not ensure that both counts are sufficiently large.

12. In an estimation procedure, what is the target parameter?

A. The single estimate produced after evaluating sample data

B. The numerical formula applied to observations from a sample

C. The unknown population quantity the procedure is designed to estimate

D. The range of plausible values reported around a sample estimate

Answer: (C)

Explanation

A target parameter is an unknown population quantity, such as $\mu$ or $p$, that the procedure seeks to estimate. A calculated sample value is an estimator or estimate rather than the population target itself.

13. What distinguishes a point estimator from an interval estimator?

A. A point estimator describes repeated samples without calculating an estimate

B. A point estimator uses sample data to produce one estimate of a population parameter

C. A point estimator identifies the parameter before any sample data are collected

D. A point estimator uses population data to produce a range of parameter values

Answer: (B)

Explanation

A point estimator is a rule or formula that converts sample data into one value estimating a population parameter. An interval estimator instead produces a range of values, which is why the two methods differ in their form of output.

14. What does a confidence interval calculate from sample data?

A. A rule for determining the size of the sampled population

B. A probability distribution for every possible sample proportion

C. A single value that replaces the target population parameter

D. A range of values that estimates a target population parameter

Answer: (D)

Explanation

A confidence interval uses sample data to calculate a range of values intended to estimate a target population parameter. A single numerical estimate is the role of a point estimator, not a confidence interval.

15. Which expression represents the general form of a confidence interval?

A. Critical value ± (point estimate)(standard error)

B. Point estimate ± (critical value)(standard error)

C. Point estimate ± (standard error)(sample size)

D. Standard error ± (critical value)(point estimate)

Answer: (B)

Explanation

A confidence interval combines a point estimate with a margin of error formed by multiplying the critical value by the standard error. The standard error reflects sampling variability, while the critical value determines the confidence level.

16. What does a 95% confidence level describe?

A. The probability that this computed interval contains the true parameter

B. The long-run proportion of intervals from the method that contain the true parameter

C. The chance that the sample estimate equals the population parameter

D. The proportion of observations lying inside this computed interval

Answer: (B)

Explanation

A 95% confidence level refers to the long-run success rate of the interval-building method across repeated samples. It does not assign a 95% probability to one particular interval after it has been calculated.

17. What is the usual effect of increasing the confidence level while keeping the data and method otherwise unchanged?

A. The standard error disappears and the interval reaches the parameter exactly

B. The interval becomes wider and its precision decreases

C. The point estimate changes and the interval width stays fixed

D. The interval becomes narrower and its precision increases

Answer: (B)

Explanation

Higher confidence requires a larger critical value, which increases the margin of error and widens the interval. The resulting interval offers greater certainty through the method but less precision because its range is broader.

18. When the population standard deviation σ is known, which formula gives a confidence interval for a population mean μ?

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

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

C. $\mu\pm z_{\alpha/2}\frac{s}{\sqrt{n}}$

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

Answer: (D)

Explanation

With known σ, the mean interval uses the sample mean, a standard-normal critical value, and the known population standard deviation divided by the square root of the sample size. The t formula applies when σ is unknown and s must be estimated from the sample.

19. Which conditions support using a z-interval for a population mean?

A. A random sample, known σ, and a sample size of at least 30

B. A representative sample, unknown σ, and a sample size below 30

C. A census, known σ, and a sample proportion near one-half

D. A random sample, estimated s, and a categorical response variable

Answer: (A)

Explanation

The z-interval for a mean is supported by random sampling, a known population standard deviation, and a sufficiently large sample, commonly taken as $n\ge30$. When σ is unknown, the corresponding mean interval uses the t-distribution and the sample standard deviation.

20. A sample has an unknown population standard deviation and contains n observations; which confidence interval should be used for the population mean?

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

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

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

D. $\mu\pm z_{\alpha/2}\frac{s}{n}$ with $df=n+1$

Answer: (B)

Explanation

When σ is unknown, the interval estimates it with s and uses the Student t critical value, with degrees of freedom $df=n-1$. The z formula requires a known population standard deviation rather than an estimate from the sample.

21. Which formula gives a confidence interval for a population proportion p?

A. $\hat{p}\pm t_{\alpha/2}\frac{\sigma}{\sqrt{n}}$

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

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

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

Answer: (C)

Explanation

A proportion interval uses the sample proportion $\hat{p}$ in its estimated standard error and applies the normal critical value. Mean intervals instead use a sample mean together with a standard deviation.

22. Which condition is required for a confidence interval for a population proportion?

A. The sample size must satisfy $n\ge30$ regardless of the expected counts

B. Both $n\hat{p}\ge15$ and $n(1-\hat{p})\ge15$ must hold

C. The sample must have equal numbers of successes and failures

D. Either $n\hat{p}\ge15$ or $n(1-\hat{p})\ge15$ must hold

Answer: (B)

Explanation

The normal approximation for a proportion interval requires sufficiently large expected counts for both successes and failures, expressed by the two inequalities. Meeting just one condition does not establish that the approximation is adequate.

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

A. The difference between the sample size and the population size

B. The total distance between the interval's lower and upper bounds

C. The probability that the sample estimate equals the population value

D. The amount added to and subtracted from a point estimate

Answer: (D)

Explanation

The margin of error quantifies estimation imprecision by extending the point estimate upward and downward to form the confidence interval. The total distance between the bounds is the interval width, which equals twice the margin of error.

24. A confidence interval is based on a point estimate of 50 and a margin of error of 4. Which expression gives the interval?

A. $4 \pm 50$

B. $50 \pm 8$

C. $50 \pm 4$

D. $50 \times 4$

Answer: (C)

Explanation

A confidence interval is expressed as the point estimate plus or minus the margin of error, giving $50 \pm 4$. Using 8 would confuse the full interval width with the margin of error.

25. With the confidence level and population variability held fixed, how can a researcher obtain a smaller margin of error?

A. Replace the point estimate with the critical value

B. Increase the sample size

C. Increase the interval width

D. Decrease the sample size

Answer: (B)

Explanation

For a fixed confidence level and variability, increasing the sample size reduces the standard error and therefore produces a smaller margin of error. Decreasing the sample size would generally increase sampling uncertainty instead.