Question

1. Which of the following statements about the t-distribution is NOT true?

1. The t-distribution is symmetric about 0.

2. The t-distribution has heavier tails than the standard normal distribution.

3. The t-distribution has support on the entire real number line

4. The t-distribution is characterized by two parameters, the mean μ and the variance σ2.

2.Which of the following statements is true?

1. When division by a factor of n-1 is used, the sample variance s2 is an unbiased estimator of the population variance σ2.

2. The t-distribution with 1 degree of freedom is equivalent to the standard normal distribution.

3. A 99% z-based confidence interval of the population mean μ based on a sample will be narrower than a 95% z-based confidence interval of the population mean μ based on the same sample.

4. If a hypothesis test is conducted at the 5% significance level, then a p-value of 0.087 would lead the researcher to reject H0.

3.For a hypothesis test of H0 : μ = 8 vs. H0 : μ > 8, the sample mean of the data is computed to be 8.24. The population standard deviation is unknown; the sample standard deviation is computed, and its value is 0.29. These sample statistics are based on a sample size of 19. It is assumed that the underlying population is normally distributed. Which of the following would be the distribution of the test statistic in this scenario?

1. The t-distribution with 8 degrees of freedom

2. The standard normal distribution

3. The t-distribution with 19 degrees of freedom

4. The t-distribution with 18 degrees of freedom

4.For a sample of 37 observations, the sample mean X̄ and the sample standard deviation s are used to construct a 95% t-based confidence interval for the population mean mu. If the confidence interval is (3.48, 4.32), then which of the following statements is NOT true?

1. The t-score used in constructing the confidence interval is 2.028.

2. The value of X̄ is 3.9.

3. The conclusion of a t-based test of H0 : μ = 3.75 vs. H1 : μ ≠ 3.75 would be to fail to reject.

4. There is a 95% probability that the true value of μ is between 3.48 and 4.32.

5.Which of the following is a valid alternative hypothesis for a one-sided hypothesis test about a population mean μ?

1. X̄ > 4.2

2. μ < 2.9

3. μ ≠ 5.4

4. μ = 3.8

Answer 1

1. Which of the following statements about the t-distribution is NOT true?

Answer:-

The t-distribution is characterized by two parameters, the mean μ, and the variance σ2.

The parameter is degrees for freedom.

2. Which of the following statements is true?

1. When division by a factor of n-1 is used, the sample variance s2 is an unbiased estimator of the population variance σ2.: - True

2. The t-distribution with 1 degree of freedom is equivalent to the standard normal distribution.: No, it Cauchy distribution.

3. A 99% z-based confidence interval of the population mean μ based on a sample will be narrower than a 95% z-based confidence interval of the population mean μ based on the same sample. No, a 99% confidence interval will be wider than a 95% confidence interval because to be more confident that the true population value falls within the interval we will need to allow more potential values within the interval.

4. If a hypothesis test is conducted at the 5% significance level, then a p-value of 0.087 would lead the researcher to reject H0. False. P-value greater than 0.05 so does not reject Ho.

Answer:- When division by a factor of n-1 is used, the sample variance s2 is an unbiased estimator of the population variance σ2.

3)

For a hypothesis test of H0 : μ = 8 vs. H0 : μ > 8, the sample mean of the data is computed to be 8.24. The population standard deviation is unknown; the sample standard deviation is computed, and its value is 0.29. These sample statistics are based on a sample size of 19. It is assumed that the underlying population is normally distributed. Which of the following would be the distribution of the test statistic in this scenario?

The t-distribution with 18 degrees of freedom

4)

For a sample of 37 observations, the sample mean X̄ and the sample standard deviation s are used to construct a 95% t-based confidence interval for the population mean mu. If the confidence interval is (3.48, 4.32), then which of the following statements is NOT true?

1. The t-score used in constructing the confidence interval is 2.028. : True

2. The value of X̄ is 3.9. : True

3. The conclusion of a t-based test of H0 : μ = 3.75 vs. H1 : μ ≠ 3.75 would be to fail to reject. : True

4. There is a 95% probability that the true value of μ is between 3.48 and 4.32. : False

Answer:- There is a 95% probability that the true value of μ is between 3.48 and 4.32.

5. Which of the following is a valid alternative hypothesis for a one-sided hypothesis test about a population mean μ?

Answer:- μ < 2.9