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Use the following information to answer the next 12 exercises: The U.S. Center for Disease Control reports that the mean life expectancy was 47.6 years for whites born in 1900 and 33.0 years for nonwhites. Suppose that you randomly survey death records for people born in 1900 in a certain county. Of the 124 whites, the mean life span was 45.3 years with a standard deviation of 12.7 years. Of the 82 nonwhites, the mean life span was 34.1 years with a standard deviation of 15.6 years. Conduct a hypothesis test to see if the mean life spans in the county were the same for whites and nonwhites. Sketch a graph of the situation. Label the horizontal axis. Mark the hypothesized difference and the sample difference. Shade the area corresponding to the p-value.

Step by step solution

01

Formulate Hypotheses

Start by establishing the null hypothesis (H0) and the alternative hypothesis (H1). The null hypothesis states that there is no difference in the mean life spans between whites and nonwhites: \( H_0: \mu_1 = \mu_2 \). The alternative hypothesis suggests that there's a difference: \( H_1: \mu_1 eq \mu_2 \).

02

Determine Test Statistic

Use the formula for the test statistic for independent samples: \[ t = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \] where \(\bar{x}_1 = 45.3\), \(\bar{x}_2 = 34.1\), \(s_1 = 12.7\), \(s_2 = 15.6\), \(n_1 = 124\), and \(n_2 = 82\). Substitute the known values into the formula.

03

Substitute Values into Test Statistic

Calculate the test statistic using the values: \[ t = \frac{(45.3 - 34.1) - 0}{\sqrt{\frac{12.7^2}{124} + \frac{15.6^2}{82}}} \]Perform the arithmetic to find the value of \( t \).

04

Calculate t-value

Compute the arithmetic from the previous step to find\[t = \frac{11.2}{\sqrt{1.30 + 2.96}} = \frac{11.2}{\sqrt{4.26}} = \frac{11.2}{2.06} \approx 5.44\].

05

Determine Degrees of Freedom

Use the formula for degrees of freedom for two independent samples: \[ df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}}\]Plug in the numbers to calculate a value for \( df \).

06

Calculate Degrees of Freedom

Compute the degrees of freedom using the formula:\[ df = \frac{(1.30 + 2.96)^2}{\frac{(1.30)^2}{123} + \frac{(2.96)^2}{81}} \approx \frac{(4.26)^2}{0.0137 + 0.108} = \frac{18.15}{0.1217} \approx 149.15 \]So, \(df \approx 149\).

07

Sketch the Distribution

Draw a normal distribution curve on the horizontal axis where the hypothesized difference (zero) is the center point. Mark the sample mean difference (11.2) and shade the tails where the calculated t-value (5.44) fits, indicating the p-value areas.

08

Conclusion from Test

Compare the t-value to a t-distribution table at \( df \approx 149\) for a two-tailed test at a significance level, commonly \( \alpha = 0.05 \). Since 5.44 is much larger than 1.976 (critical t-value), 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.

t-distribution

When diving into hypothesis testing, especially in small sample sizes, we encounter the **t-distribution**. This statistical concept is a cousin of the normal distribution but is a bit broader at the tails. Why does it matter? Because many real-world samples are small and don't perfectly match the normal distribution curve.

The t-distribution comes into play when dealing with sample data rather than population data. It accounts for the extra variability that might arise with smaller samples. When we calculate a t-value (a statistic that reflects how our sample data compares to the null hypothesis), we refer to the t-distribution to determine the likeliness of our observed data.

• It is useful when the sample size is under 30.
• As the sample size grows, the t-distribution resembles the normal distribution.
• This distribution is vital for calculating probabilities (p-values) associated with our t-test results.

Two-Sample t-Test

The **Two-Sample t-Test** is a valuable statistical test used to determine if there is a significant difference between the means of two independent groups. Think of it as a way to compare two separate sets of observations to see if one group significantly differs from another.

Conducting a two-sample t-test involves several key steps:

• Formulate the hypotheses: Here, the null hypothesis states there is no difference between group means, while the alternative posits a difference.
• Compute the test statistic: This involves measuring the difference between group means and dividing it by the standard error, derived from each group's variance and sample size.
• Calculate the p-value: This tells us the probability that the observed data would occur if the null hypothesis were true.
• Make a decision: Compare the test statistic to a critical value from the t-distribution. If the test statistic is beyond this critical value, the null hypothesis is rejected.

By examining the differences in life spans between whites and nonwhites from our exercise, the two-sample t-test helps us see if observed disparities are statistically significant.

Degrees of Freedom

**Degrees of Freedom** is a somewhat abstract concept but plays an important role in statistical calculations such as t-tests. It represents the number of values in a calculation that are free to vary. Simply put, it's related to the amount of information we have.

• In a two-sample t-test, the degrees of freedom help determine the shape of the t-distribution curve used to reference critical values.
• For our two-sample scenario, the degrees of freedom are calculated using a specific formula that considers both sample sizes and their variances.
• In our example with lifespans, degrees of freedom were calculated to be approximately 149. This value is used when deciding whether to accept or reject our null hypothesis.

The degrees of freedom impact the t-distribution's shape, with more degrees allowing for a closer approximation to the normal distribution. Thus, understanding degrees of freedom enhances our insight into the reliability and precision of our test conclusions.