Question

Question 4

Consider the following hypotheses:

H0: mean = 5

H1: mean < 5

A test is performed with a sample of size 25. The sample mean was 4.67 and the population standard deviation is 1.2. Assume that the population is approximately normal. Use the TI-84 PLUS calculator to compute the P-value.

Question 5

Consider the following hypotheses:

H0: mean = 9

H1: mean > 9

A test is performed with a sample of size 49. The sample mean was 12.8 and the population standard deviation is 14. Assume that the population is approximately normal. Use the TI-84 PLUS calculator to compute the P-value.

Round your answer to four decimal places (for example: 0.0138). Write only a number as your answer.

Question 7

Consider the following hypotheses:

H0: mean = 16

H1: mean 16

A test is performed with a sample of size 100. The sample mean was 6.09 and the population standard deviation is 60. Assume that the population is approximately normal. Use the TI-84 PLUS calculator to compute the P-value.

Round your answer to four decimal places (for example: 0.0138). Write only a number as your answer.

Answer 1

In the above problems, we are to test if the unknown population means are greater than, less than or different from the hypothesized population mean. Thus our null hypothesis in every case becomes testing whether "mu", the unknown population mean is equal to the hypothesized value of the population mean and the alternative hypothesis becomes H1a: mu < mu0,

H1b: mu > mu0 and H1c: mu not equal to mu0.

For testing each of the above hypothesis, our test statistic is: t=(xbar - mu0)/(s/sqrt(n)) ; where xbar = sample mean weight, mu0 = hypothesized value of the population mean, s = sample standard deviation, n = no. of observations in the sample.

We reject H0, if t(observed) > tau(alpha), for the greater case, or if t(observed) < - tau(alpha) for the lesser case or if absolute value of t(observed) is greater than tau(alpha/2), where tau(alpha) is the upper alpha point of a Standard normal distribution, or if the p-value is less than the level of significance (alpha).

4) The required p-value is = 0.0846 (Thus we do not reject H0 and conclude that there is not sufficient evidence to conclude that the population mean is less than 5)

5) The required p-value is = 0.0287 (Thus we reject H0 and conclude that there is sufficient evidence to conclude that the population mean is greater than 9)

7) The required p-value is = 0.0986 (Thus we do not reject H0 and conclude that there is not sufficient evidence to conclude that the population mean is different from 16)

(All the answers are rounded to 4 decimal places, obtained keeping level of significance (alpha) = 0.05, and are obtained using R-software, code and output are attached below for verification.)