MATLAB: An Introduction with Applications
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ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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Transcribed Image Text:### Hypothesis Testing on Mean Daily High Temperature
The National Weather Service states that the mean daily high temperature for October in a large Midwestern city is \( 57^\circ F \). A local weather service suspects that this value is not accurate and wants to perform a hypothesis test to determine whether the mean is actually lower than \( 57^\circ F \).
### Given Data
- Sample Mean (\(\bar{X}\)): \( 55^\circ F \)
- Population Standard Deviation (\(\sigma\)): \( 5.7^\circ F \)
- Significance Level (\(\alpha\)): 0.01
We want to perform the hypothesis test at the \(\alpha = 0.01\) significance level.
### Hypothesis Statements
1. Greater than \(57^\circ F\):
- Null Hypothesis (\(H_0\)): \( \mu = 57^\circ F \)
- Alternative Hypothesis (\(H_a\)): \( \mu > 57^\circ F \)
- Test statistics: \( z = -1.92 \)
- Critical Value: \( z = -2.33 \)
- Conclusion: Fail to reject \(H_0: \mu = 57^\circ F\). There is not sufficient evidence to support the claim that the mean is greater than \(57^\circ F\).
2. Less than \(57^\circ F\):
- Null Hypothesis (\(H_0\)): \( \mu = 57^\circ F \)
- Alternative Hypothesis (\(H_a\)): \( \mu < 57^\circ F \)
- Test statistics: \( z = -1.92 \)
- Critical Value: \( z = -2.33 \)
- Conclusion: Fail to reject \(H_0: \mu = 57^\circ F\). There is not sufficient evidence to support the claim that the mean is less than \(57^\circ F\).
### Summary
For both hypothesis tests, the test statistics provided (\(z = -1.92\)) is compared against the critical value (\(z = -2.33\)). In both cases, the null hypothesis is not rejected, indicating insufficient evidence to support either of the alternative hypotheses regarding the mean daily high temperatures differing from \( 57^\circ F \).
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