Flashcards: Sampling Distributions and Confidence Intervals — 57 cards

What is a population parameter?

A numerical descriptive measure of a population.

Why is a population parameter usually unknown?

Because it is based on all population observations.

What is a sample statistic?

A numerical descriptive measure calculated from sample observations.

What symbol denotes the population mean?

The population mean is denoted by μ.

What symbol denotes the sample mean?

The sample mean is denoted by x̄.

What symbol denotes the population proportion?

The population proportion is denoted by p.

What symbol denotes the sample proportion?

The sample proportion is denoted by p̂.

What is the sampling distribution of a statistic?

It is the probability distribution of that statistic across all samples of size n.

What is the mean of the sampling distribution of the mean?

It equals the population mean μ.

How is the standard deviation of the sampling distribution of the mean calculated?

It is $\frac{\sigma}{\sqrt{n}}$, called the standard error.

What is the population mean in the ESCP weight example?

63 kg.

What is the population standard deviation in the ESCP weight example?

23 kg.

What is the sample size in the ESCP weight example?

100 students.

What is the standard error in the ESCP weight example?

2.3 kg.

What is the sampling distribution of the mean if the original variable is normal?

It is normally distributed for any sample size.

What does the Central Limit Theorem say about the sampling distribution of the mean?

It can be approximated by a normal model.

How does sample size affect the Central Limit Theorem's approximation?

The approximation improves as sample size increases.

When is sample size usually large enough for the Central Limit Theorem?

When n is at least 30.

What does the required sample size for the Central Limit Theorem depend on?

It depends on the shape of the sampled distribution.

What is the standard deviation of the sampling distribution for samples of size 50?

It is approximately 1.41 cm.

What is the population mean height in the example with sample size 50?

The population mean height is 172 cm.

What is the probability of a sample mean of at least 184 cm for samples of size 50?

The probability is approximately 0.

What is the z-score for a sample mean of 184 cm in samples of size 50?

The z-score is 8.48.

What is the z-score for a sample mean of 140 pounds in the BMI example?

The z-score is −1.03.

What is the probability of a sample mean of 140 pounds in the BMI example?

The probability is 15.15%.

Is a sample mean of 140 pounds in the BMI example considered exceptionally unusual?

No, it is low but not exceptionally unusual.

What is the sampling distribution of a proportion?

It is the probability distribution of p̂ across all samples of size n.

When is the sampling distribution of p̂ approximately normal?

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

What is the standard error of p̂ for 52% support and sample size 400?

The standard error is 0.025.

How is the standard error of p̂ calculated for 52% support and sample size 400?

By $\sqrt{(0.52)(0.48)/400}=0.025$.

What is the target parameter in statistical estimation?

The unknown population parameter an estimation procedure aims to estimate.

What does a point estimator use to calculate an estimate?

Sample data.

What does a point estimator calculate?

A single estimate of a population parameter.

What is a confidence interval in statistics?

A formula using sample data to calculate a range estimating a population parameter.

What is the general confidence-interval formula?

Point Estimate ± (Critical Value)(Standard Error)

What does the confidence level represent in statistics?

The long-run proportion of intervals containing the true parameter.

What proportion of intervals contain the true parameter in a 95% confidence procedure?

95% of intervals contain the true parameter.

What proportion of intervals miss the true parameter in a 95% confidence procedure?

5% of intervals miss the true parameter.

What effect does increasing the confidence level have on the interval width?

It requires a wider interval.

What trade-off results from increasing the confidence level?

Greater certainty produces lower precision.

What is the confidence interval formula for μ when σ is known?

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

What conditions are required for the z-interval for μ?

Random sample, known σ, and sample size n≥30 for normality.

What is the confidence interval formula for μ when σ is unknown?

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

How is the degrees of freedom calculated for the t-interval?

Degrees of freedom equal $n-1$.

How does the Student t-distribution compare to the standard normal distribution?

It is bell-shaped, symmetric, with fatter tails than normal.

What happens to the Student t-distribution as sample size n increases?

It approaches the standard normal distribution.

What is the formula for a confidence interval for a population proportion?

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

What sample conditions are required for a confidence interval for p?

A random or representative sample and sufficiently large expected counts are required.

What are the minimum expected counts for a confidence interval for p?

$n\hat{p}\ge15$ and $n(1-\hat{p})\ge15$

What was the sample proportion of voters satisfied with Bertrand Delanoë in the 2010 IPSOS survey?

53% of 1011 voters were satisfied.

What was the 95% confidence interval for satisfaction with Bertrand Delanoë in 2010?

From 49.9% to 56.1%.

What is the margin of error in statistics?

It is the amount added to and subtracted from the point estimate to form a confidence interval.

How is a confidence interval expressed using margin of error?

As Point Estimate plus or minus the margin of error.

What is the formula for margin of error (ME)?

ME equals Critical Value times Standard Error.

What happens to sample size when margin of error decreases at fixed confidence and variability?

The sample size must increase.

Which confidence levels are most commonly used?

90%, 95%, and 99%.

What are the normal critical values for 90%, 95%, and 99% confidence levels?

1.645, 1.96, and 2.576 respectively.