V. Is your conclusion in part iii. for the Apparatus Finals consistent with your conclusion for the Individual All-Around Finals from Part A? Explain. V. Consider the requirements for a t-test for difference of population means and determine if it was reasonable to proceed with the t-test in part iii. (Hint: Consider the sample sizes for female gymnasts and male gymnasts. Then, consider the normal quantile plots for female floor scores and male floor scores give
ONLY ANSWER THE BOLDED QUESTIONS ON THE BOTTOM (IV AND V)
Part A: Individual All-Around Finals We will first look at the floor exercise scores in the Individual Female All-Around Finals and in the Individual Male All-Around Finals.
iii. Construct a 95% confidence interval for the difference in population
Sample Mean x1 = 12.687
Sample Standard Deviation s1 = 1.477
Sample Mean x1 = 12.174
Sample Standard Deviation s1 = 0.944
Sample Size n1 = 104
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df = 208
tcritical = 1.971 (found using excel)
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Sp2 = (106 - 1)1.4772 + (104 - 1)0.9442 / (106 - 1) + (104 - 1)
Sp2 = 1.543
E = 1.9711.543106+1.543104
E = 0.338
CI = (12.687 - 12.174) + 0.338, (12.687 - 12.174) - 0.338
CI = (0.851, 0.175)
Therefore, the 96% confidence interval for the difference between the population means, μ1 - μ2 , is 0.851 < μ1 - μ2 < 0.175.
iv. Using the confidence interval in part iii., can we infer at the 5% significance level that the population mean floor exercise score in the Individual All-Around Finals is different for female gymnasts and male gymnastics?
The null and alternative hypothesis are:
μ1 represents the population mean for the Female All-Around Finals Score.
μ2 represents the population mean for the Male All-Around Finals Score.
H0: μ1 = μ2
H1: μ1 ≠ μ2
Two-tailed test, t-test for population means, two independent samples, and unknown population standard deviations.
t = (12.687 - 12.174) / 1.543106+1.543104
t = (0.513) / 1.543106+1.543104
t = 2.992
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Test statistic ≤ Critical value of t → 2.992 > 1.971
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P-value = 0.003 (found using excel)
P-value > significance level → 0.003 < 0.05
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It can be inferred that at the 5% significance level the population mean floor exercise score in the Individual All-Around Finals is different for female gymnasts and male gymnastics
It is concluded that the null hypothesis H0 is rejected. Therefore, there is not sufficient evidence to claim that the population mean floor exercise score in the Individual All-Around Finals is different for female gymnasts and male gymnastics.
Part B: Apparatus Finals Let’s now look at the floor exercise scores in the Female Apparatus Finals and in the Male Apparatus Finals.
iii. Based on statistical data, can we infer at the 5% significance level that female gymnasts have a higher population mean floor exercise score in the Apparatus Finals than male gymnasts? (Hint: Use the results of i. and ii. and apply the t-test for difference of population means).
Sample Mean x1 = 13.181
Sample Standard Deviation s1 = 2.971
Sample Size n1 = 24
Sample Mean x1 = 13.031
Sample Standard Deviation s1 = 0.714
Sample Size n1 = 24
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The null and alternative hypothesis are:
μ1 represents the population mean for the Female Apparatus Finals Score.
μ2 represents the population mean for the Male Apparatus Finals Score.
H0: μ1 = μ2
H1: μ1 > μ2
Since, the alternative hypothesis contains a greater than symbol, it is a right tailed test.
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Significance level: α = 0.05
df = n1 + n2 - 2
df = 24 + 24 - 2
df = 46
tcritical = tα, df
= t0.05, 46
= 1.679 (found using t-table)
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Sp2 = (24 - 1)2.9712 + (24 - 1)0.7142 / (24 - 1) + (24 - 1)
Sp2 = (23)2.9712 + (23)0.7142 / 23 + 23
Sp2 = (23)2.9712 + (23)0.7142 / 46
Sp2 = 4.668
t = (13.181 - 13.031) / 4.66824+4.66824
t = 0.15 / 4.66824+4.66824
t = 0.15 / 0.624
t = 0.240
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Test statistic ≤ Critical value of t → 0.240 < 1.679
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P-value = 0.811 (found using excel)
P-value > significance level → 0.811 > 0.05
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It is concluded that the null hypothesis H0 is not rejected. There is not sufficient evidence to support the claim that the population mean μ1 is Higher than μ2 at the 0.05 significance level.
IV. Is your conclusion in part iii. for the Apparatus Finals consistent with your conclusion for the Individual All-Around Finals from Part A? Explain.
V. Consider the requirements for a t-test for difference of population means and determine if it was reasonable to proceed with the t-test in part iii. (Hint: Consider the sample sizes for female gymnasts and male gymnasts. Then, consider the normal quantile plots for female floor scores and male floor scores given below. You can read more about normal quantile plots in 6.5 Assessing Normality in the textbook.)
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