Question

Fill out the answer sheet for each of the hypothesis tests below, using a 5% level of significance. Use 4 decimal places, unless the p-value is really small (less than 0.0001), in which case use at least 2 significant figures This means 2 non-zero numbers somewhere after the decimal point). Axis of graphs should be drawn with a ruler or done on a computer.

Suppose that students own an average of 4 pairs of jeans. 8 people from your class were surveyed to determine if the average for students at De Anza College is higher than 4.

DATA TO USE: 2,2,3,4,6,6,8,9

a. Give the null and alternative hypotheses: Ho: _______________ Ha: ___________________

b. In words, CLEARLY state what your random variable X or P' represents.

c. State the distribution to use for the test. If t, include the degrees of freedom. If normal, include the mean and standard deviation.

d. p-value = ______________

e. In 1 – 2 complete sentences, explain what the p-value means for this problem.

f. Use the previous information to draw a graph of this situation. CLEARLY, label and scale the horizontal axis and shade the region(s) corresponding to the p-value. The values of your sample statistic and the hypothesized value of the population parameter should be on the axis.

g. Indicate the correct decision (“reject the null hypothesis” or “do not reject the null hypothesis”) and write an appropriate conclusion, using COMPLETE SENTENCES.

Decision:

Conclusion:

h. Construct a 95% Confidence Interval for the true mean or proportion. Include a sketch of the graph of the

situation. Label the point estimate and the lower and upper bounds of the Confidence Interval.

Confidence Interval: ( ___________________ , ___________________ )

i. Interpret the confidence interval in a complete sentence.

Answer 1

a) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ = 44

Ha: μ > 44

This corresponds to a right-tailed test, for which a t-test for one mean, with unknown population standard deviation will be used.

b) X is random variable

(c) Rejection Region: Based on the information provided, the significance level is α=0.05,degrees of freedom= n-1=8-1=7 and the critical value for a right-tailed test is tc​=1.895.

The rejection region for this right-tailed test is R=t:t>1.895

Test Statistics

The t-statistic is computed as follows:

d) Using the P-value approach: The p-value is p=0.1623

e) Since p=0.1623≥0.05, it is concluded that the null hypothesis is not rejected.

f)

G) Decision: DO NOT REJECT NULL HYPOTHESIS H0.

Conclusion: It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean μ is greater than 4, at the 0.05 significance level.

h) t critical value = 2.36
s_M = √(2.67²/8) = 0.94
μ = M ± t(s_M)
μ = 5 ± 2.36*0.94
μ = 5 ± 2.2

95% CI [2.8, 7.2].

i) You can be 95% confident that the population mean (μ) falls between 2.8 and 7.2