Questions: Professional Golfers' Earnings Two random samples of earnings of professional golfers were selected. One sample was a=0.05, is there a difference in the means? The data are in thousands of dollars. Use the critical value method with tables. PGA 446, 1147, 9188, 10,508, 4910 8553, 7573, 375 LPGA 48, 76, 122, 466, 863 1876, 2029, 4364, 2921 Use μ₁ for the mean earnings of PGA golfers. Assume the variables are normally distributed and the variances are unequal. State the hypotheses and identify the claim with the correct hypothesis. H₀: not claim claim This hypothesis test is a two-tailed test. = =, ≠ ≠, > <, < μ₁, μ₂ ×, 0

Professional Golfers' Earnings Two random samples of earnings of professional golfers were selected. One sample was a=0.05, is there a difference in the means? The data are in thousands of dollars. Use the critical value method with tables.

PGA
446, 1147, 9188, 10,508, 4910
8553, 7573, 375

LPGA
48, 76, 122, 466, 863
1876, 2029, 4364, 2921

Use μ₁ for the mean earnings of PGA golfers. Assume the variables are normally distributed and the variances are unequal.

State the hypotheses and identify the claim with the correct hypothesis.
H₀: not claim claim
This hypothesis test is a two-tailed test.

= =, ≠ ≠, > <, < μ₁, μ₂
×, 0

Transcript text: Professional Golfers' Earnings Two random samples of earnings of professional golfers were selected. One sample was $a=0.05$, is there a difference in the means? The data are in thousands of dollars. Use the critical value method with tables. \begin{tabular}{lrrrr} PGA & & & \\ \hline 446 & 1147 & 9188 & 10,508 & 4910 \\ 8553 & 7573 & 375 & & \\ & & & & \\ \hline LPGA & & & & \\ \hline 48 & 76 & 122 & 466 & 863 \\ 1876 & 2029 & 4364 & 2921 & \end{tabular} Use $\mu_{1}$ for the mean earnings of PGA golfers. Assume the variables are normally distributed and the variances are unequal. State the hypotheses and identify the claim with the correct hypothesis. $H_{0}$ : $\square$ not claim claim $\qquad$ This hypothesis test is a two-tailed $\square$ test. \begin{tabular}{|c|c|c|} \hline$\square=\square$ & $\square \neq \square$ & $\square>\square$ \\ $\square<\square$ & $\mu_{1}$ & $\mu_{2}$ \\ \hline$\times$ & 0 \\ \hline \end{tabular}

Solution

Solution Steps

Step 1: State the Hypotheses

We are testing whether there is a difference in the means of earnings between PGA and LPGA golfers. The hypotheses are stated as follows:

Null Hypothesis ($H_0$): $\mu_1 = \mu_2$ (There is no difference in means)
Alternative Hypothesis ($H_a$): $\mu_1 \neq \mu_2$ (There is a difference in means)

This is a two-tailed test.

Step 2: Calculate the Standard Error

The standard error ($SE$) is calculated using the formula:

\[ SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} = \sqrt{\frac{17599078.0}{8} + \frac{2261602.25}{9}} = 1565.6225 \]

Step 3: Calculate the Test Statistic

The test statistic ($t$) is calculated as follows:

\[ t = \frac{\bar{x}_1 - \bar{x}_2}{SE} = \frac{5337.5 - 1418.3333}{1565.6225} = 2.5033 \]

Step 4: Calculate the Degrees of Freedom

The degrees of freedom ($df$) for Welch's t-test is calculated using the formula:

\[ df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}} = \frac{6008253433570.682}{699249409384.3777} = 8.5924 \]

Step 5: Calculate the p-value

The p-value is calculated as follows:

\[ P = 2(1 - T(|t|)) = 2(1 - T(2.5033)) = 0.0348 \]

Step 6: Determine the Critical Value

For a significance level of $\alpha = 0.05$ and $df \approx 8.5924$, the critical value from the t-distribution table is approximately:

\[ \text{Critical value} = 2.2786 \]

Step 7: Conclusion

Since the calculated $p$-value $0.0348$ is less than the significance level $\alpha = 0.05$, we reject the null hypothesis. This indicates that there is a statistically significant difference in the means of earnings between PGA and LPGA golfers.