Question

According to a recent census, 15% of the people in the United States are of Hispanic origin. One county supervisor believes her county has a lower proportion of Hispanic people than the nation as a whole (p < 0.15). She looks at their most recent survey data, which was a random sample of 485 county residents, and found that 43 of those surveyed are of Hispanic origin. Test statistic Z is found to be -3.78, calculate p-value.

Answer 1

Answer:

The P-value of the given test is 0.00008.

Step-by-step explanation:

We are given that according to a recent census, 15% of the people in the United States are of Hispanic origin.

She looks at their most recent survey data, which was a random sample of 485 county residents, and found that 43 of those surveyed are of Hispanic origin.

Let p = proportion of Hispanic people than the nation as a whole.

So, Null Hypothesis, $H_0$ : p = 0.15      {means that county has a 15% proportion of Hispanic people than the nation as a whole}

Alternate Hypothesis, $H_A$ : p < 0.15     {means that county has a lower proportion of Hispanic people than the nation as a whole}

The test statistics that would be used here One-sample z-test for proportions;

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

where, $\hat p$ = sample proportion of Hispanic origin people = $\frac{43}{485}$ = 0.09

           n = sample of county residents = 485

So, the test statistics  =  $\frac{0.09-0.15}{\sqrt{\frac{0.15(1-0.15)}{485} } }$

                                     =  -3.78

The value of z test statistic is -3.78.

Now, P-value of the test statistics is given by the following formula;

            P-value = P(Z < -3.78) = 1 - P(Z $\leq$ 3.78)

                          = 1 - 0.99992 = 0.00008

Answer 2

The p-value of the given hypothesis test for the recent census is; 0.00008

What is the p-value of the statistic?

Let us define the hypothesis;

Null Hypothesis: p = 0.15

Alternate Hypothesis: p < 0.15  

Let us calculate the sample proportion;

p^ = x/n = 43/485 = 0.09

Formula for z-score is;

z = (p^ - p)/√(p(1 - p)/n)

z = (0.09 - 0.15)/√(0.15(1 - 0.15)/485)

z = -3.78

From online p-value from z-score calculator, we have;

p-value = 0.00008

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