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
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Solution

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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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