Question

Use the following information to complete steps ​(a) through ​(d) below. A random sample of size n 1 equals 31 results in a sample mean of 123.3 and a sample standard deviation of 8.5 . An independent sample of size n 2 equals 50 results in a sample mean of 130.8 and sample standard deviation of 7.3 . Does this constitute sufficient evidence to conclude that the population means differ at the alpha equals 0.01 level of​ significance? ​(a) What type of test should be​ used? A. A hypothesis test regarding the difference of two means using a​ matched-pairs design. B. A hypothesis test regarding the difference between two population proportions from independent samples. C. A hypothesis test regarding two population standard deviations. D. A hypothesis test regarding the difference of two means using​ Welch's approximate t. ​(b) Determine the null and alternative hypotheses. Choose the correct answer below. A. Upper H 0 : mu 1 less than or equals mu 2 ​; Upper H 1 : mu 1 not equals mu 2 B. Upper H 0 : mu 1 equals mu 2 ​; Upper H 1 : mu 1 greater than mu 2 C. Upper H 0 : mu 1 equals mu 2 ​; Upper H 1 : mu 1 not equals mu 2 D. Upper H 0 : mu 1 equals mu 2 ​; Upper H 1 : mu 1 less than or equals mu 2 ​(c) Use technology to calculate the​ P-value. nothing ​(Round to three decimal places as​ needed.) ​(d) Draw a conclusion based on the hypothesis test. Choose the correct answer below. A. There is not sufficient evidence to reject the null hypothesis because the ​P-valuegreater than alpha . B. There is sufficient evidence to reject the null hypothesis because the ​P-valueless than alpha . C. There is not sufficient evidence to reject the null hypothesis because the ​P-valueless than alpha . D. There is sufficient evidence to reject the null hypothesis because the ​P-valuegreater than alpha .

Answer 1

Solution:

(a) What type of test should be​ used?

Answer: D. A hypothesis test regarding the difference of two means using​ Welch's approximate t.​

Explanation: The samples are independent and we would like to compare the two means from these two populations, therefore, the appropriate test would be Welch's approximate t.

​(b) Determine the null and alternative hypotheses.

Answer: The null and alternative hypotheses are:

(c) Use technology to calculate the​ P-value. ​(Round to three decimal places as​ needed.)

Answer: We can use the TI-84 calculator to find the p-value. The steps are as follows:

Press STAT and then scroll right to TESTS

Scroll down to 2-SampTTest and scroll right to Stats.

Enter the values as:

Press calculate and the output is:

Therefore, the p-value is:

​(d) Draw a conclusion based on the hypothesis test. Choose the correct answer below.

B. There is sufficient evidence to reject the null hypothesis because the ​P-value is less than alpha.