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Education influences attitude and lifestyle. Differences in education are a big factor in the "generation gap." Is the younger generation really better educated? Large surveys of people age 65 and older were taken in \(n_{1}=32\) U.S. cities. The sample mean for these cities showed that \(\bar{x}_{1}=15.2 \%\) of the older adults had attended college. Large surveys of young adults (age \(25-34\) ) were taken in \(n_{2}=35\) U.S. cities. The sample mean for these cities showed that \(\bar{x}_{2}=19.7 \%\) of the young adults had attended college. From previous studies, it is known that \(\sigma_{1}=7.2 \%\) and \(\sigma_{2}=5.2 \%\) (Reference: American Generations, S. Mitchell). Does this information indicate that the population mean percentage of young adults who attended college is higher? Use \(\alpha=0.05\).

Step by step solution

01

Define Hypotheses

To determine if the population mean percentage of young adults who attended college is higher, we set two hypotheses:- Null hypothesis: \( H_0: \mu_1 = \mu_2 \) (There is no difference in the mean percentages).- Alternative hypothesis: \( H_a: \mu_1 < \mu_2 \) (The mean percentage of young adults is higher). We use a one-tail test since we want to determine if the younger generation is more educated.

02

Calculate the Standard Error

The standard error (SE) of the difference between the two sample means is calculated using:\[ SE = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} \]Substitute \(\sigma_1 = 7.2\%\), \(n_1 = 32\), \(\sigma_2 = 5.2\%\), and \(n_2 = 35\):\[ SE = \sqrt{\frac{7.2^2}{32} + \frac{5.2^2}{35}} \]

03

Perform the Calculation for the Standard Error

Calculating each part separately:- \(\frac{7.2^2}{32} = 1.62\)- \(\frac{5.2^2}{35} = 0.77\)Thus, the standard error is:\[ SE = \sqrt{1.62 + 0.77} = \sqrt{2.39} \approx 1.546 \]

04

Calculate the Test Statistic

The test statistic \( z \) is calculated as:\[ z = \frac{\bar{x}_2 - \bar{x}_1}{SE} \]\[ z = \frac{19.7\% - 15.2\%}{1.546} = \frac{4.5}{1.546} \approx 2.91 \]

05

Determine Critical Value and Make a Decision

For a significance level of \(\alpha = 0.05\), the critical value for a one-tailed test is approximately \(1.645\). Since our calculated \( z \) value of \(2.91\) is greater than \(1.645\), we reject the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

In hypothesis testing, the null hypothesis is a statement about the population that implies no change or no effect. It's denoted as \( H_0 \), and in many cases, it represents a statement of equality such as "there is no difference" or "there is no effect".

In our given situation, the null hypothesis is \( H_0: \mu_1 = \mu_2 \). This means that we assume there is no difference between the percentage of older adults and young adults who have attended college.

It's important to note that we approach hypothesis testing with the assumption that the null hypothesis is true. We gather data and perform calculations to see if we have enough evidence to reject this assumption. If the result allows us to conclude significantly, then the null hypothesis is rejected.

Alternative Hypothesis

The alternative hypothesis is the statement that we aim to support through our hypothesis test. It specifies a difference or effect that we suspect exists in the population. It is denoted as \( H_a \) or sometimes \( H_1 \).

In the exercise, the alternative hypothesis is \( H_a: \mu_1 < \mu_2 \). This suggests that the percentage of young adults attending college (\( \mu_2 \)) is higher than that of older adults (\( \mu_1 \)).

The alternative hypothesis is key because it defines the direction of the test. Since our hypothesis is directional ("less than"), we are conducting a one-tail test. This impacts our calculations and the critical value we use to make our decision.

Standard Error

Standard error (SE) is a measure used to quantify the amount of variation or "spread" in a set of sample means we get if we repeated the sampling process multiple times. It's calculated for the difference between two sample means, and it helps to determine how typical the observed difference is under the null hypothesis.

The formula for standard error of the difference between two means is:
\[ SE = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} \]

Here, \( \sigma_1 \) and \( \sigma_2 \) are the standard deviations of the two populations, and \( n_1 \) and \( n_2 \) are the sample sizes. For our case:

• \( \sigma_1 = 7.2\% \) for the older adults.
• \( \sigma_2 = 5.2\% \) for the young adults.

Given these values, we calculated the SE to be approximately \( 1.546 \). This is a crucial step before we move on to calculate the test statistic.

Test Statistic

The test statistic is a standardized value that is calculated from sample data during a hypothesis test. It is used to determine how far your sample statistic is from the null hypothesis, under the assumption that the null hypothesis is true. For the comparison of two means, it's often calculated using the formula:
\[ z = \frac{\bar{x}_2 - \bar{x}_1}{SE} \]

Where \( \bar{x}_2 \) and \( \bar{x}_1 \) are the sample means, and \( SE \) is the standard error.

In our situation, with a calculated SE of \( 1.546 \):

• \( \bar{x}_1 = 15.2\% \) for older adults.
• \( \bar{x}_2 = 19.7\% \) for young adults.

The calculated test statistic \( z \) is approximately \( 2.91 \).

This statistic is then compared to a critical value determined by our significance level \( \alpha \). It helps decide whether to reject the null hypothesis.

One-Tail Test

A one-tail test is a hypothesis test in which the region of rejection is on only one side of the sampling distribution. It's used when we want to determine if there is either an increase or decrease, but not both. In our study, we are checking if young adults have a higher college attendance rate, so we are only considering the possibility of an increase.

For a significance level of \( \alpha = 0.05 \), the critical value for a one-tail test is approximately \( 1.645 \).

Since the calculated test statistic \( z = 2.91 \) is greater than the critical value, this indicates enough evidence to reject the null hypothesis. In simpler terms, the data suggests that a higher percentage of young adults have attended college compared to older adults.