Question

10. For a particular scenario, we wish to test the hypothesis H₀ : p = 0.52. For a sample of size 50, the sample proportion p̂ is 0.42. Compute the value of the test statistic z_obs. (Express your answer as a decimal rounded to two decimal places.)

4. 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?

a		The standard normal distribution
b		The t-distribution with 18 degrees of freedom
c		The t-distribution with 19 degrees of freedom
d		The t-distribution with 8 degrees of freedom

6. 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?

	a	The value of X̄ is 3.9.
	b	The t-score used in constructing the confidence interval is 2.028.
	c	There is a 95% probability that the true value of μ is between 3.48 and 4.32.
	d	The conclusion of a t-based test of H0 : μ = 3.75 vs. H1 : μ ≠ 3.75 would be to fail to reject.

Answer 1

10. Zobs=(-p)/(√p(1-p)/n)

=(0.42-0.52)/(√0.52*0.48/50)= -1.415

So required z statistic is -1.415

4. Since population SD is unknown, for the test of the true mean value here, we would use a tn-1 distribution. That is a t test with df: n-1= 19-1= 18

So option b is correct.

6. Confidence interval is given as:

Sample mean +-Margin of error

So the sample mean = (4.32+3.48)/2 = 3.9

So option a is correct

t-score should be t0.025,n-1 ie t0.025,36= 2.028

So option b is correct.

We can also see that u=3.75 is contained in the CI above. So option d is also correct, we fail to Reject H0.

Now option c is false. From the CI we can only say that we are 95% confident that the true mean is contained in the interval above, we cannot say the probability is 0.95

So option c is false.