Question

Results from previous studies showed 76% of all high school seniors from a certain city plan to attend college after graduation. A random sample of 200 high school seniors from this city reveals that 162 plan to attend college. Does this indicate that the percentage has increased from that of previous studies? Test at the 5% level of significance.

Answer 1

Answer:

Yes, this indicate that the percentage has increased from that of previous studies.

Step-by-step explanation:

We are given that Results from previous studies showed 76% of all high school seniors from a certain city plan to attend college after graduation.

A random sample of 200 high school seniors from this city reveals that 162 plan to attend college.

Let p = % of all high school seniors from a certain city who plan to attend college after graduation

SO, Null Hypothesis, $H_0$ : p $\leq$ 76%   {means that the percentage has not  increased from that of previous studies}

Alternate Hypothesis, $H_a$ : p > 76%   {means that the percentage has increased from that of previous studies}

The test statistics that will be used here is One-sample z proportion statistics;

              T.S.  = $\frac{\hat p-p}{{\sqrt{\frac{\hat p(1-\hat p)}{n} } } } }$  ~ N(0,1)

where,  $\hat p$ = sample proportion of high school seniors from this city who  

                   plan to attend college = $\frac{162}{200}$  = 0.81

            n = sample of high school seniors = 200

So, test statistics  =  $\frac{0.81-0.76}{{\sqrt{\frac{0.81(1-0.81)}{200} } } } }$  

                               =  1.8025

Now at 5% significance level, the z table gives critical value of 1.6449 for right-tailed test. Since our test statistics is more than the critical value of z so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region.

Therefore, we conclude that the percentage has increased from that of previous studies.