Question

The Institute of Education Sciences measures the high school dropout rate as the percentage of 16- through 24-year-olds who are not enrolled in school and have not earned a high school credential. In 2009, the high school dropout rate was 8.1%. A polling company recently took a survey of 1000 people between the ages of 16 and 24 and found 6.5% of them are high school dropouts. The polling company would like to determine whether the proportion of dropouts has changed from the historical value of 0.081. Use the 5% significance level. Set up the null and alternative hypotheses, calculate the Test Statistic, p-value, and write your conclusion in a sentence

Answer 1

Answer:

We conclude that the proportion of dropouts has changed from the historical value of 0.081.

Step-by-step explanation:

We are given that in 2009, the high school dropout rate was 8.1%. A polling company recently took a survey of 1000 people between the ages of 16 and 24 and found 6.5% of them are high school dropouts.

The polling company would like to determine whether the proportion of dropouts has changed from the historical value of 0.081.

Let p = proportion of school dropouts rate

SO, Null Hypothesis, $H_0$ : p = 0.081   {means that the proportion of dropouts has not changed from the historical value of 0.081}

Alternate Hypothesis, $H_A$ : p $\neq$ 0.081   {means that the proportion of dropouts has changed from the historical value of 0.081}

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 dropout rate = 6.5%

            n = sample of people = 1000

So, test statistics  =   $\frac{0.065-0.081}{{\sqrt{\frac{0.065(1-0.065)}{1000} } } } }$ 

                               =  -2.05

Also, P-value is given by the following formula;

         P-value = P(Z < -2.05) = 1 - P(Z $\leq$ 2.05)

                                              = 1 - 0.97982 = 0.0202 or 2.02%

Now at 5% significance level, the z table gives critical values between -1.96 and 1.96 for two-tailed test. Since our test statistics does not lies within the range of critical values 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 proportion of dropouts has changed from the historical value of 0.081.