Question

1). Conduct the appropriate hypothesis test with the following information, and provide your p-value as your final answer.  The null hypothesis states that H₀:= 250 versus the alternative H_a: > 250. The sample of size 26 resulted in a mean of 255.0 and a standard deviation of 8.79.

Find the p-value in this example.

2). Conduct the appropriate hypothesis test with the following information, and provide your p-value as your final answer.  The null hypothesis states that H₀: = 0.75 versus the alternative H_a:  0.75.  The sample proportion is 0.72; this statistic was calculated using a sample of size 222.

Find the p-value in this example.

3). Consider the following hypotheses, H₀:= 0.5 versus the alternative H_a:  < 0.5. Suppose that the appropriate hypothesis test is conducted, and the test results in a p-value of 0.2356.  What would be the p-value of this test if the alternative hypothesis had instead been H_a: not equal to 0.5?

Find the p-value in this example.

4). A statistician carries out a significance test of the null hypothesis that the proportion wanting to reduce gluten at the university is the same as for all adults (i.e. the same as the general population) versus the alternative hypothesis that a smaller proportion of students would say they want to reduce or be free of gluten.

What are the null and alternative hypotheses in this situation?

1. H_o:  = 0.30 H_a:  < 0.30
2. H_o:  = 0.30 H_a:  > 0.30
3. H_o:  = 0.26 H_a:  < 0.26
4. H_o: = 0.26 H_a:  > 0.26

Answer 1

1) The test statistic t = ()/(s/)

= (255 - 250)/(8.79/)

= 2.9

For alpha = 0.05, the critical value is t^* = 1.708

Since the test statistic value is greater than the critical value (2.9 > 1.708), so we should not reject H₀.

P-value = P(T > 2.9)

= 1 - P(T < 2.9)

= 1 - 0.9962 = 0.0038

B) The test statistic z = ( - p)/sqrt(p(1 - p)/n)

= (0.72 - 0.75)/sqrt(0.75 " (1 - 0.75)/222)

= -1. 03

At alpha = 0.05, the critical values are +/- z_0.025 = +/- 1.96

Since the test statistic value is not less than the lower critical value (-1.03 > -1.96), so we should not reject H₀.

P-value = 2 * P(Z < -1.03)

= 2 * 0.1515 = 0.3030

3) P-value = 2 * 0.2356 = 0.4712

4) Option - c) H₀: p = 0.26

H_a: p < 0.26