Question

The Washington Post reported that 80% of all high school students had computers at their home. A random sample of 500 high school students in an urban area is taken, and it reveals that 425 students have computers at home. Test if the proportion of high school students in this urban area who have computers at their home exceeds 80%. Use a 0.1 level of significance.

Part A: State the null and alternative hypotheses.

a. H0: p = 0.8, Ha: p (does not =) 0.8

b. H0: p > 0.8, Ha: p = 0.8

c. H0: p = 0.8, Ha: p > 0.8

d. H0: p (does not =) 0.8, Ha: p < 0.8

Part B: Find the value of test statistic.

a. 0.85

b. -0.85

c. 2.795

d. -2.795

Part C: Find the rejection region.

a. Z > 1.96 and Z < -1.96

b. Z > 1.96

c. Z > 2.326

d. Z > 1.282

Part D: What is your decision, reject H0 or not?

a. Do not reject H0, because the value of test statistic is within the rejection region.

b. Reject H0, because the value of test statistic is not within the rejection region.

c. Reject H0, because the value of test statistic is within the rejection region.

d. Do not reject H0, because the value of test statistic is not within the rejection region.

Part E: Interpret the decision in the context. At 0.10 level of significance, …

a. There is sufficient evidence that the proportion of high school students in this area who have computers at their home exceeds 80%.

b. There is insufficient evidence that the proportion of high school students in this area who have computers at their home is not 80%.

c. There is insufficient evidence that the proportion of high school students in this area who have computers at their home exceeds 80%.

d. There is sufficient evidence that the proportion of high school students in this area who have computers at their home is not 80%.

Answer 1

We want to test hypothesis Ho p 0.8 Ha: p>0.8 This is an upper tailed test Null Hypothesis, Ho p 0.8 is also correct) (Alternative Hypothesis, also called H1 T425 n 500 Sample proportion, p - 0.85, n -500 and claimed proportion. P = 0.8 Significance level, a 0.1 now, calculating statistics P × (1-P) 0.8 × (1-0.8) 500 Standard Deviation of p, ơp 0.0179 p-0.8 (0.85)-0.8 0.0179 2.795085 心observed Test statistic,20-2795 Critical value ze za 20.120.1 1.282 (From z table , using interpolation, 5/6 th distance between 1.28 and 1.29) Rejection Criteria: Reject Ho if zo > 1.282 d. Z > 1.282 Rejection region has been shaded 0.1 1.282 Decision: Since zo > e, we reject the null hypothesis Hence at 10% significance level, there is sufficient evidence to conclude that p is greater than to 0.8, i.e, the proportion of high school students in this area who have computers at their home exceeds 80%