Question 1

Statisticians have shown that the mathematical process that derived the Student t distribution is robust, which means that if the sampled population is nonnormal, the t-test of the population mean  is still valid, provided that the population is not extremely nonnormal.

Accepted Answer

The t-test is robust to nonnormality as long as the sample size is sufficiently large and the nonnormality is not extreme. This is a well-established statistical concept.

Question 2

If a sample of size 20 is randomly selected from a population, the value of A for the probability P(-A  t  A) = 0.95 is 2.093.

Accepted Answer

This statement is true according to the standard normal distribution table. When the probability of the left tail is 0.025 and the sample size is 20, the corresponding value of A is 2.093.

Question 3

A random sample of size 9 produced a sample mean equal to 13.5 and a standard deviation of 3.2. The margin of error associated with a 95% confidence interval estimate for the population mean is approximately 2.46.

Accepted Answer

The formula for margin of error is (z-score)*(standard deviation of sample/ sqrt(sample size)).

Here, we are given the following information:
Sample size (n) = 9
Sample mean (x̄) = 13.5
Standard deviation (s) = 3.2
Confidence level = 95%
Since the sample size is less than 30 and the population standard deviation is unknown, we need to use the t-distribution to find the margin of error.

The critical t-value for a 95% confidence interval with 8 degrees of freedom (9-1) is approximately 2.306. Therefore, the margin of error is:
2.306 * (3.2 / √9) = 2.4584 (rounded to two decimal places)

This is approximately 2.46, as given in the question. Therefore, the statement is true, and the answer is A.

Question 4

With all other factors held constant, confidence intervals constructed with small samples tend to have greater margin of error than those constructed from larger samples.

Accepted Answer

This is because smaller samples provide less information about the population and may not be as representative, leading to a wider range of possible values for the parameter of interest.

Question 5

If a sample has 10 observations and a 90% confidence estimate for  is needed, the appropriate t-score is 1.833.

Accepted Answer

This is correct according to a t-distribution table with 9 degrees of freedom and a 90% confidence level.

Question 6

If a sample has 15 observations and a 95% confidence estimate for  need to be constructed, the appropriate t-score is 2.131.

Accepted Answer

The answer of If a sample has 15 observations and...

Question 7

A random sample of size 10 produced a sample mean equal to 12 and a standard deviation equal to .15. Based on this information, the upper limit for a 95% confidence interval estimate is approximately 12.107.

Accepted Answer

The upper limit for a 95% confidence interval estimate with a sample size of 10, sample mean of 12, and standard deviation of 0.15 is 12.107, therefore the statement is true.

Question 8

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

Accepted Answer

The answer of If a sample has 18 observations and...

Question 9

The t-distribution approaches the normal distribution as the number of degrees of freedom increases.

Accepted Answer

As the number of degrees of freedom increases, the t-distribution becomes narrower and its shape approaches that of the standard normal distribution. In fact, when the degrees of freedom are very large, the t-distribution is almost identical to the normal distribution.

Question 10

A random sample of size 10 produced a sample mean equal to 12 and a standard deviation equal to .15. Based on this information, the margin of error associated with a 90% confidence interval estimate for the population mean is 1.8331.

Accepted Answer

The formula for the margin of error for a 90% confidence interval is:
margin of error = 1.645 * (standard deviation/square root of sample size)
Using the given information:
margin of error = 1.645 * (.15/sqrt(10)) = 0.0812
Therefore, the statement that the margin of error is 1.8331 is false.

Question 11

The t-distribution is used to construct confidence interval for the population mean  when the population is unknown and the sample size is small.

Accepted Answer

The answer of The t-distribution is used to construct confidence...

Question 12

Student's t-distribution is a sampling distribution for a random variable, t, derived from a normally distributed population, that is (1) single-peaked above the random variable's mean, median, and mode of zero, (2) perfectly symmetrical about this central value, and (3) characterized by tails extending indefinitely in both directions from the center, approaching, but never touching, the horizontal axis.

Accepted Answer

Student's t-distribution has these characteristics: it is single-peaked and symmetrical around a mean of zero, with tails that extend indefinitely but never touch the horizontal axis, indicating the distribution of the t-statistic under the assumption of a normally distributed population.

Question 13

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

Accepted Answer

The answer of In order to determine the p-value associated...

Question 14

The "shape" of the t-distribution changes as the value of the sample mean changes.

Accepted Answer

The shape of the t-distribution is affected by the degrees of freedom (related to sample size), not the value of the sample mean.

Question 15

If a sample has 15 observations and a 90% confidence estimate for  is needed, the appropriate t-score is 1.341.

Accepted Answer

The appropriate t-score for a 90% confidence interval with 15 observations is actually 1.753, not 1.341.

Question 16

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 4.52 seconds to 5.18 seconds.

Accepted Answer

The given information does not provide any clear indication of whether the stated confidence interval is true or false. The confidence interval may or may not be accurate based on the sample size, distribution, and assumptions.

Question 17

Regardless of the degrees of freedom, every t-distribution is symmetric around zero.

Accepted Answer

This is true. The symmetry of the t-distribution is a fundamental property and does not depend on the number of degrees of freedom.

Question 18

A 90% confidence interval estimate for the population mean constructed with a small sample will have a margin of error that is approximately  90% of the population size n.

Accepted Answer

The margin of error for a confidence interval estimate depends on the sample size and standard deviation, not the population size. Therefore, the statement is false.

Question 19

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...

Question 20

If a sample of size 15 is randomly selected from a population, the value of A for the probability P(t  A) = 0.01 is 2.602.

Accepted Answer

The value of A for the given probability and sample size can be found using a t-table or a t-distribution calculator. Using both methods, we get A = 2.947. Therefore, the statement is false as the calculated value of A is different from the given value of 2.602.

Quiz 10: Inference From Small Samples

Statisticians have shown that the mathematical process that derived the Student t distribution is robust, which means that if the sampled population is nonnormal, the t-test of the population mean is still valid, provided that the population is not extremely nonnormal.

If a sample of size 20 is randomly selected from a population, the value of A for the probability P(-A t A) = 0.95 is 2.093.

A random sample of size 9 produced a sample mean equal to 13.5 and a standard deviation of 3.2. The margin of error associated with a 95% confidence interval estimate for the population mean is approximately 2.46.

With all other factors held constant, confidence intervals constructed with small samples tend to have greater margin of error than those constructed from larger samples.

If a sample has 10 observations and a 90% confidence estimate for is needed, the appropriate t-score is 1.833.

If a sample has 15 observations and a 95% confidence estimate for need to be constructed, the appropriate t-score is 2.131.

A random sample of size 10 produced a sample mean equal to 12 and a standard deviation equal to .15. Based on this information, the upper limit for a 95% confidence interval estimate is approximately 12.107.

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

The t-distribution approaches the normal distribution as the number of degrees of freedom increases.

A random sample of size 10 produced a sample mean equal to 12 and a standard deviation equal to .15. Based on this information, the margin of error associated with a 90% confidence interval estimate for the population mean is 1.8331.

The t-distribution is used to construct confidence interval for the population mean when the population is unknown and the sample size is small.

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

The "shape" of the t-distribution changes as the value of the sample mean changes.

If a sample has 15 observations and a 90% confidence estimate for is needed, the appropriate t-score is 1.341.

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 4.52 seconds to 5.18 seconds.

Regardless of the degrees of freedom, every t-distribution is symmetric around zero.

A 90% confidence interval estimate for the population mean constructed with a small sample will have a margin of error that is approximately 90% of the population size n.

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

If a sample of size 15 is randomly selected from a population, the value of A for the probability P(t A) = 0.01 is 2.602.