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[size=2.25]Based on the national weather report, the temperature in California...
Based on the national weather report, the temperature in California has a mean of 53° F with a standard deviation of 4.6° F. However, the Agriculture Weather Services claims that the mean temperature in California is more than that of the average of 53°F based on the farmers' reports. In a random sample of 15 farmers, the mean temperature was 57.2° F. A right-tailed hypothesis test was conducted, which resulted in a 0.0371 p-value.
A.)Were the results of this study statistically significant at the 0.05 alpha level? Explain why or why not. B.)Compute Cohen's d. C.) Interpret the value of Cohen's d that you computed in question 8. D.) What are two different changes that could be made to increase the statistical power of this research study? Answer & Explanation
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A. Since the p-value is less than the alpha level or 0.0371 < 0.05, then we will reject the null hypothesis. Since we reject Ho, we can conclude that the results are statistically significant at 0.05 alpha level. This means that there is enough evidence to support Agriculture Weather Services' claim that the true mean temperature is more than 53°F.
B. Cohen's d = 0.913
C. The effect size is large since Cohen's d value is greater than 0.8.
D. Increasing the sample size. This is the most practical way of increasing test power. Increasing the sample size will provide us with more information about the population. Increasing the significance level. Power refers to the probability of avoiding a type 2 error. If we increase the significance level, we will obtain a lower probability of committing a type 2 error. In that case, the power of the study will increase.
[size=2.25]Step-by-step explanation
A. Were the results statistically significant? In this research, the easiest way to determine whether the result is statistically significant is to use the P-value approach. In this approach, we will compare the p-value of the test from the alpha level. P-value approach: Reject Ho if p-value < alpha. - If we reject Ho, then the results are statistically significant.
- If we fail to reject Ho, then the results are not statistically significant.
Given alpha level = 0.05 p-value = 0.0371 Decision Since the p-value is less than the alpha level or 0.0371 < 0.05, then we will reject the null hypothesis. Since we reject Ho, we can conclude that the results are statistically significant at 0.05 alpha level. This means that there is enough evidence to support Agriculture Weather Services' claim that the true mean temperature is more than 53°F.
B. Cohen's d. - It is a way to measure the effect size.
- P-value detects whether there is an effect while Cohen's d tells how large the effect is.
For One Mean T-test, the formula for Cohen's d is: Cohen's d = sXˉ−μ X̄ = sample mean μ = hypothesized mean (population mean) s = sample standard deviation Given sample mean X̄ = 57.2°F hypothesized mean μ = 53°F sample standard deviation s = 4.6°F Solution Cohen's d = sXˉ−μ= 4.657.2−53 = 0.913
C. Interpretation of Cohen's d. The rule of thumb in interpreting Cohen's d is provided below. No effect = 0 Small effect = 0.2 Medium or moderate effect = 0.5 Large effect = 0.8
The effect size is large since Cohen's d value is greater than 0.8. This effect size indicates that the observed sample mean and the population mean differ by 0.913 standard deviations of the given data.
D. Increasing Statistical Power There are different ways to increase the statistical power of a study. - Increase the sample size.
- Increase the significance level.
- Switch from two-tailed test to one-tailed.
- Increase the difference between the means.
- Improve the process to obtain a smaller standard deviation.
For the given research study, the changes that we can make to increase the statistical power are: - Increasing the sample size. This is the most practical way of increasing test power. Increasing the sample size will provide us with more information about the population. This will result in a more accurate (powerful) result.
- Increasing the significance level. Power refers to the probability of avoiding a type 2 error. If we increase the significance level, we will obtain a lower probability of committing a type 2 error. In that case, the power of the study will increase.
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