Questions: A city wants to advertise itself as having a highly educated population. The proportion of the city's population with a bachelor's degree or higher is 37%. The city believes that 37% is significantly higher than the general proportion of the United States population with a bachelor's degree or higher of 33.4%. The sample statistic is Pick The population parameter is Pick

A city wants to advertise itself as having a highly educated population. The proportion of the city's population with a bachelor's degree or higher is 37%. The city believes that 37% is significantly higher than the general proportion of the United States population with a bachelor's degree or higher of 33.4%.

The sample statistic is Pick
The population parameter is Pick

Transcript text: A city wants to advertise itself as having a highly educated population. The proportion of the city's population with a bachelor's degree or higher is $37 \%$. The city believes that $37 \%$ is significantly higher than the general proportion of the United States population with a bachelor's degree or higher of $33.4 \%$. The sample statistic is Pick $\square$ The population parameter is $\square$ Pick

Solution

Solution Steps

Step 1: Identify the Sample Statistic and Population Parameter

The sample statistic is the proportion of the city's population with a bachelor's degree or higher, which is $ \hat{p} = 0.37 $.

The population parameter is the proportion of the United States population with a bachelor's degree or higher, which is $ p_0 = 0.334 $.

Step 2: Formulate the Hypotheses

We want to test if the city's proportion is significantly higher than the national proportion. Therefore, we set up the hypotheses as follows:

Null Hypothesis ($ H_0 $): $ p = 0.334 $
Alternative Hypothesis ($ H_a $): $ p > 0.334 $

Step 3: Calculate the Test Statistic

The test statistic for a proportion is calculated using the formula:

\[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]

Substituting the given values:

\[ Z = \frac{0.37 - 0.334}{\sqrt{\frac{0.334 \times (1 - 0.334)}{1000}}} = 2.4137 \]

Step 4: Determine the P-value

The p-value associated with the test statistic $ Z = 2.4137 $ is $ 0.0079 $.

Step 5: Define the Critical Region

For a significance level of $ \alpha = 0.05 $ and a one-tailed test, the critical region is $ Z > 1.6449 $.

Step 6: Make a Decision

Since the test statistic $ Z = 2.4137 $ falls in the critical region ($ Z > 1.6449 $) and the p-value $ 0.0079 $ is less than $ \alpha = 0.05 $, we reject the null hypothesis.

Final Answer

The city's proportion of $ 37\% $ is significantly higher than the national proportion of $ 33.4\% $. Therefore, the city can claim that its population is more highly educated than the general population of the United States.

$\boxed{\text{Reject } H_0}$

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