Question

A decade-old study found that the proportion, P, of high school seniors who believed that "getting rich" was an important personal goal was 75%. A researcher decides to test whether or not that percentage still stands. He finds that, among the 215 high school seniors in his random sample, 170 believe that "getting rich" is an important goal. Can he conclude, at the 0.1 level of significance, that the proportion has indeed changed?

Perform a two-tailed test.

The null hypothesis:
The alternative hypothesis:
The type of test statistic (list degrees of freedom if necessary):
The value of the test statistic: (Round to at least three decimal places):
the p-value:

Can we conclude that the proportion of high school seniors who believe that "getting rich" is an important goal has changed? Yes or no

Answer 1

Null hypothesis: H0; P = 0.75

Alternative hypothesis: Ha; P ≠ 0.75

z = -2.55

P value = P(Z<-2.556) + P(Z>2.556) = 0.011

Decision: We FAIL to REJECT the null hypothesis

Can we conclude that the proportion of high school seniors who believe that "getting rich" is an important goal has changed = NO

Rule

If;

P-value > significance level --- accept Null hypothesis

P-value < significance level --- reject Null hypothesis

Z score > Z(at 99% confidence interval) ---- reject Null hypothesis

Z score < Z(at 99% confidence interval) ------ accept Null hypothesis

Explanation:

Given;

n= 250 represent the random sample taken

Null hypothesis: H0; P = 0.75

Alternative hypothesis: Ha ≠ 0.75

Test statistic z score can be calculated with the formula below;

z = (p^−po)/√{po(1−po)/n}

Where,

z= Test statistics

n = Sample size = 250

po = Null hypothesized value = 0.75

p^ = Observed proportion = 170/250 = 068

Substituting the values we have

z = (0.68-0.75)/√(0.75(1-0.75)/250)

z = −2.556

To determine the p value (test statistic) at 0.01 significance level, using a two tailed hypothesis.

P value = P(Z<-2.556) + P(Z>2.556) = 0.011

Since z at 0.01 significance level is between -2.58 and +2.58 and the z score for the test (z = -2.556) which doesn't falls with the region bounded by Z at 0.01 significance level. And also the two-tailed hypothesis P-value is 0.011 which is lower than 0.01. Then we can conclude that we don't have enough evidence to reject the null hypothesis, and we can say that at 1% significance level the null hypothesis is valid, therefore we fail to reject the null hypothesis.