Question 1

If the sampled population is nonnormal, the t -test of the population mean m is still valid, provided that the condition is not extreme.

Accepted Answer

The Central Limit Theorem suggests that as the sample size increases, the distribution of the sample mean approaches a normal distribution, even if the population from which the sample is drawn is not normally distributed. Therefore, for sufficiently large sample sizes, the t-test can still be valid for non-normal populations, provided the deviation from normality is not extreme.

Question 2

The statistic s² is an unbiased and consistent estimator of s ² .

Accepted Answer

The notation seems to be a typographical error or a placeholder, but in statistical theory, an estimator is considered unbiased if its expected value equals the parameter it estimates, and it is consistent if it converges in probability to the true parameter value as the sample size increases. Without specific details on what "s²" represents, the statement is technically true in the context of statistical properties of estimators.

Question 3

The t- distribution is used in a confidence interval for a mean when the actual standard error is not known.

Accepted Answer

The t-distribution is used in situations where the population standard deviation is unknown and must be estimated from the sample data. It is used to calculate confidence intervals for the population mean.

Question 4

In forming a 95% confidence interval for a population mean from a sample size of 20, the number of degrees of freedom from the t -distribution equals 20.

Accepted Answer

The number of degrees of freedom for a t-distribution with a sample size of 20 is actually 19. The formula for calculating the degrees of freedom is sample size minus 1, which in this case would be 20-1 = 19.

Question 5

In estimating the population mean with the population standard deviation unknown, if the sample size is 16, there are 8 degrees of freedom.

Accepted Answer

The formula for degrees of freedom in this case is n-1, where n is the sample size. Therefore, in this case there would be 15 degrees of freedom, not 8.

Question 6

If a sample has 15 observations and a 95% confidence estimate for m is needed, the appropriate value of t is 1.753.

Accepted Answer

For a sample of 15 observations (n=15) and a 95% confidence level, the degrees of freedom (df) would be n-1, which equals 14. The appropriate t-value for 14 degrees of freedom and a 95% confidence level is typically around 2.145, not 1.753.

Question 7

In order to determine the p -value associated with hypothesis testing about the population mean m , it is necessary to know the value of the test statistic.

Accepted Answer

The p-value in hypothesis testing is calculated based on the test statistic, which measures how far the sample statistic is from the null hypothesis value under the assumption that the null hypothesis is true.

Question 8

The t- distribution assumes that the population is normally distributed.

Accepted Answer

The answer of The t- distribution assumes that the population...

Question 9

The statistic  when the sampled population is normal is Student t -distributed with n degrees of freedom.

Accepted Answer

The answer of The statistic  when the sampled population...

Question 10

The t -distribution is used to develop a confidence interval estimate of the population mean when the population standard deviation is unknown.

Accepted Answer

This is correct, as the t-distribution is specifically designed to handle situations where the population standard deviation is unknown and has to be estimated from the sample data.

Question 11

The chi-squared distribution can be used in constructing confidence intervals and carrying out hypothesis tests regarding the value of a population variance.

Accepted Answer

This statement is true. The chi-squared distribution is commonly used in statistical inference for population variance. Confidence intervals and hypothesis tests can be constructed using the chi-squared distribution.

Question 12

The test statistic used to test hypotheses about the population variance is given by c ² = ( n - 1)s ² \ s ², which is chi-squared distributed with n - 1 degrees of freedom when the population is normally distributed with variance equal to s ² .

Accepted Answer

The test statistic for testing hypotheses about a population variance, $(n - 1)s^2 / \sigma^2$, follows a chi-squared distribution with $n - 1$ degrees of freedom, assuming the population from which the sample is drawn is normally distributed.

Question 13

A race car driver tested his car for time from 0 to 60 mph, and in 20 tests obtained an average of 48.5 seconds with a standard deviation of 1.47 seconds. A 95% confidence interval for the 0 to 60 time is 45.2 seconds to 51.8 seconds.

Accepted Answer

The given information does not provide enough details to calculate a confidence interval directly, and the stated interval does not mathematically follow from the provided average and standard deviation without knowing the sample size and assuming a normal distribution or applying the central limit theorem for large samples.

Question 14

If a sample has 18 observations and a 90% confidence estimate for m is needed, the appropriate value of t is 1.740.

Accepted Answer

The t-value for a 90% confidence interval with 18 degrees of freedom is 1.740.

Question 15

The area to the right of a certain critical value on a chi-squared distribution is 0.025. For 5 degrees of freedom, the critical value is 11.143.

Accepted Answer

The answer of The area to the right of a...

Question 16

A random sample is drawn from a normal distribution with mean m and variance s ² . The random variable ( n - 1)S² \ s ² has a chi-squared probability distribution with n degrees of freedom.

Accepted Answer

The random variable $(n-1)S^2 / \sigma^2$ has a chi-squared distribution with $n-1$ degrees of freedom, not $n$ degrees of freedom.

Question 17

In order to interpret the p -value associated with hypothesis testing about the population mean m , it is necessary to know the value of the test statistic.

Accepted Answer

The p-value can be interpreted without knowing the exact value of the test statistic; it indicates the probability of observing data at least as extreme as the observed data under the null hypothesis.

Question 18

A condition of using the chi-squared test statistic for testing the variance is that the population random variable have a normal distribution.

Accepted Answer

The chi-squared test for variance assumes that the data are sampled from a population that follows a normal distribution, as the test statistic is derived under this assumption.

Question 19

The t- distribution allows the calculation of confidence intervals for means for small samples when the population variance is not known, regardless of the shape of the distribution in the population.

Accepted Answer

The t-distribution is used for calculating confidence intervals for means for small samples when the population variance is unknown, but it assumes that the underlying population distribution is approximately normal.

Question 20

Two uses for the population variance are to measure risk and consistency.

Accepted Answer

The population variance measures the spread of a population's data, which is a measure of risk. It also measures the consistency of the data since a smaller variance indicates that the data is more tightly clustered around the mean.

Quiz 11: Inference About a Population