Question 1

A market researcher computed a confidence interval for a population proportion using a 95% confidence level.Her boss decided that she wanted a 99% confidence level instead.The new interval with 99% confidence level will be wider than the original one with a 95% confidence level.

Accepted Answer

Increasing the confidence level from 95% to 99% requires capturing more of the population proportion to be more certain, thus widening the confidence interval.

Question 2

If the population is normal and its standard deviation, σ, is known and the sample size, n, is large (n ≥ 30), interval estimates for the population mean must be determined using z-values.

Accepted Answer

When the population standard deviation (σ) is known and the sample size is large (n ≥ 30), the Central Limit Theorem allows us to use the z-distribution for constructing interval estimates for the population mean, regardless of the population's original distribution.

Question 3

In determining the interval estimates for a population proportion using the sample proportion, it is appropriate to use the z-distribution.

Accepted Answer

When determining interval estimates for a population proportion using the sample proportion, it is appropriate to use the z-distribution, especially when the sample size is large (typically n>30) and the conditions of np and n(1-p) being greater than or equal to 10 are met, ensuring the sampling distribution of the sample proportion is approximately normal.

Question 4

If the population is not normal but its standard deviation, σ is known and the sample size, n is large (n ≥ 30), z-distribution values may be used to determine interval estimates for the population mean.

Accepted Answer

According to the Central Limit Theorem, when the sample size is large (n ≥ 30), the sampling distribution of the sample mean will be approximately normally distributed regardless of the shape of the population distribution. Therefore, z-distribution values can be used for interval estimates of the population mean when the population standard deviation is known, even if the population is not normally distributed.

Question 5

A t-distribution is similar to a normal distribution, but with flatter tails.

Accepted Answer

A t-distribution has heavier (not flatter) tails than a normal distribution, which accounts for the increased variability when estimating population parameters from small sample sizes.

Question 6

In determining the interval estimates for a population variance using the sample variance, it is appropriate to use the values from a chi-square distribution rather than a t-distribution.

Accepted Answer

The chi-square distribution is used for interval estimates of a population variance because the sample variance follows a chi-square distribution when the population is normally distributed, not a t-distribution which is used for means.

Question 7

If the degrees of freedom in a t distribution increase, the difference between the t values and the z values will also increase.

Accepted Answer

As the degrees of freedom in a t distribution increase, the t distribution approaches the normal distribution, making the t values closer to the z values, not further apart.

Question 8

When the population standard deviation, σ, is unknown the sample standard deviation, s, is used in determining the interval estimate for the population mean.

Accepted Answer

When the population standard deviation (σ) is unknown, the sample standard deviation (s) is used as an estimate for σ in calculating the interval estimate for the population mean, typically in the context of constructing confidence intervals.

Question 9

Use of the chi-square statistic to estimate the population variance is extremely robust to the assumption that the population is normally distributed.

Accepted Answer

The chi-square statistic for estimating population variance is sensitive to the assumption of normality in the underlying population. If the population is not normally distributed, the chi-square test may not be appropriate or reliable.

Question 10

In order to find values in the t distribution table, you must determine the appropriate degrees of freedom based on the sample sizes.

Accepted Answer

The degrees of freedom for a t distribution, which are crucial for finding values in the t distribution table, are determined based on the sample size(s) involved in the statistical analysis.

Question 11

If the population is normal and its standard deviation, σ, is known but the sample size is small, z-distribution values may not be used to determine interval estimates for the population mean.

Accepted Answer

When the population is normal and the standard deviation is known, z-distribution values can be used to determine interval estimates for the population mean regardless of the sample size.

Question 12

When a statistic calculated from sample data is used to estimate a population parameter, it is called a point estimate.

Accepted Answer

A point estimate is indeed a statistic calculated from sample data that is used to estimate the value of a population parameter.

Question 13

An assumption underlying the use of t-statistic in sample-based estimation is that the population is normally distributed.

Accepted Answer

The t-statistic is based on the assumption that the underlying population from which the sample is drawn is normally distributed, especially important when dealing with small sample sizes.

Question 14

Suppose a random sample of 16 is selected from a population with a normal distribution with a known population standard deviation σ of 10.Assume that the sample mean is 4.2.Based on a 90% confidence interval for the population mean, we can conclude that 0.1 is a plausible number for the population mean μ.

Accepted Answer

The confidence interval is 4.2 ± 1.729(10/√16) = (2.771, 5.629). Since 0.1 is not within this interval, it is not a plausible number for the population mean μ.

Question 15

Catherine Chao, Director of Marketing Research, is evaluating consumer acceptance of a new toothpaste package.Her staff reports that 17% of a random sample of 200 households prefers the new package to all other package designs.If Catherine concludes that 17% of all households prefer the new package, she is using _______.

Accepted Answer

Catherine is using a point estimate because she is using the specific percentage (17%) from the sample to estimate the preference for the new package in the entire population.

Question 16

Like a t-distribution, a chi-square distribution is symmetrical and extends from minus infinity to plus infinity.

Accepted Answer

A chi-square distribution is not symmetrical and only extends from 0 to plus infinity because it is based on squared values, making negative values impossible.

Question 17

When a range of values is used to estimate a population parameter, it is called a range estimate.

Accepted Answer

When a range of values is used to estimate a population parameter, it is called a confidence interval.

Question 18

Given the error we are willing to tolerate, the sample size is determined by the mean, µ of the population and the confidence level.

Accepted Answer

The sample size is primarily determined by the desired margin of error, the standard deviation (or an estimate of it), and the confidence level. The mean of the population does not directly determine the sample size.

Question 19

You are thinking of using a t-table to find a 95 percent confidence interval for the mean μ of a population.The distribution of the population is normal and the population standard deviation is unknown.A random sample of size n is drawn from this population.You may use the t-distribution only if the sample size n is small.

Accepted Answer

The t-distribution can be used for any sample size when the population standard deviation is unknown, but it is especially useful for small sample sizes. For large samples, the t-distribution approaches the normal distribution.

Question 20

In estimating the sample size necessary to estimate a population mean, the error of estimation, E, is equal to the difference between the sample mean and the sample standard deviation.

Accepted Answer

The error of estimation, E, is typically defined as the maximum difference expected between the sample mean and the population mean, not the difference between the sample mean and the sample standard deviation.

Quiz 8: Statistical Inference: Estimation for Single Populations