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A researcher takes sample temperatures in Fahrenheit for 17 days from Sacramento (Group 1) and 18 days from Orlando (Group 2). Test the claim that the mean temperature in Sacramento is greater than the mean temperature in Orlando. Use a significance level of [tex]\alpha = 0.10[/tex]. Assume the populations are approximately normally distributed with unequal variances. You obtain the following two samples of data. Round answers to 4 decimal places.

[tex]
\[
\begin{array}{|c|c|}
\hline
\text{Sacramento} & \text{Orlando} \\
\hline
66.9 & 37.4 \\
\hline
60.3 & 52.3 \\
\hline
48.5 & 46.6 \\
\hline
51.1 & 50.7 \\
\hline
62.3 & 52.0 \\
\hline
55.6 & 69.0 \\
\hline
58.1 & 32.5 \\
\hline
65.1 & 45.4 \\
\hline
72.7 & 45.9 \\
\hline
60.9 & 40.8 \\
\hline
64.8 & 51.2 \\
\hline
44.1 & 59.7 \\
\hline
64.8 & 33.9 \\
\hline
75.0 & 55.4 \\
\hline
64.1 & 53.7 \\
\hline
49.3 & 44.0 \\
\hline
69.9 & 40.5 \\
\hline
& 37.4 \\
\hline
\end{array}
\]
[/tex]

Answer :

To test the claim that the mean temperature in Sacramento is greater than the mean temperature in Orlando, we can perform a two-sample t-test for unequal variances. Here's a detailed step-by-step solution:

1. State the Hypotheses:

- Null Hypothesis ([tex]\(H_0\)[/tex]): The mean temperature in Sacramento ([tex]\(\mu_1\)[/tex]) is equal to the mean temperature in Orlando ([tex]\(\mu_2\)[/tex]), i.e., [tex]\(\mu_1 = \mu_2\)[/tex].
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): The mean temperature in Sacramento is greater than the mean temperature in Orlando, i.e., [tex]\(\mu_1 > \mu_2\)[/tex].

2. Collect the Sample Data:

- Sacramento temperatures: 66.9, 60.3, 48.5, 51.1, 62.3, 55.6, 58.1, 65.1, 72.7, 60.9, 64.8, 44.1, 64.8, 75, 64.1, 49.3, 69.9
- Orlando temperatures: 37.4, 52.3, 46.6, 50.7, 52, 69, 32.5, 45.4, 45.9, 40.8, 51.2, 59.7, 33.9, 55.4, 53.7, 44, 40.5, 37.4

3. Calculate Sample Means and Standard Deviations:

- Mean of Sacramento temperatures: 60.7941
- Mean of Orlando temperatures: 47.1333
- Standard deviation of Sacramento temperatures: 8.725
- Standard deviation of Orlando temperatures: 9.4079

4. Conduct a Two-Sample t-test (with unequal variances):

Using a t-test for unequal variances (also known as Welch's t-test), we determine the test statistic and p-value:

- Test Statistic: 4.4568
- p-value: 0.0

5. Decision and Conclusion:

We compare the p-value to the significance level ([tex]\(\alpha = 0.10\)[/tex]). Since the p-value (0.0) is less than [tex]\(\alpha\)[/tex], we reject the null hypothesis.

Conclusion: There is significant evidence at the 0.10 significance level to support the claim that the mean temperature in Sacramento is greater than the mean temperature in Orlando.

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