8.4 Testing a Claim about Two Population Means 317 our answer to the last part o
ID: 3200711 • Letter: 8
Question
8.4 Testing a Claim about Two Population Means 317 our answer to the last part of the any more precise than what ("What is the P-value for this test?") can't be we have said already: the P-value is less than 0.005 Section 8.4 Exercises Exercise 8.19 Researchers at the Centerville Health Department selected a random sample of 35 adult women in Centerville and recorded their resting pulse rates. The mean pulse rate for the women was 71.5 beats per minute and the sample standard deviation was 3.2 beats per minute. The same researchers selected a sample of 33 adult men in Centerville and recorded their resting pulse rates. The mean resting pulse rate for the men was 68.5 beats per minute and the sample standard deviation for the men was 3.7 beats per minute.Explanation / Answer
Solution :-
8.19 )
The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below:
State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: 1 - 2 = 0
Alternative hypothesis: 1 - 2 0
Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the difference between sample means is too big or if it is too small.
Formulate an analysis plan. For this analysis, the significance level is 0.10. Using sample data, we will conduct a two-sample t-test of the null hypothesis.
Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).
SE = sqrt[(s12/n1) + (s22/n2)]
SE = sqrt[((3.2)2/35) + ((3.7)2/33] = sqrt(0.2926 + 0.4148) = sqrt(0.7075) = 0.8411
DF = (s12/n1 + s22/n2)2 / { [ (s12 / n1)2 / (n1 - 1) ] + [ (s22 / n2)2 / (n2 - 1) ] }
DF = ((3.2)2/35 + (3.7)2/33)2 / { [ ((3.2)2 / 35)2 / (34) ] + [ ((3.7)2 / 33)2 / (32) ] }
DF = (0.2926 + 0.7075)2 / { [ (0.2926)2 / (34) ] + [ (0.7075)2 / (32) ] }
= 1.0002 / (0.0025 + 0.0156) = 1.0002 / 0.0181 = 5.5259
t = [ (x1 - x2) - d ] / SE = [ (71.5 - 68.5) - 0 ] / 3.51 = -3/0.8411
= - 3. 567
where s1 is the standard deviation of sample 1, s2 is the standard deviation of sample 2, n1 is the size of sample 1, n2 is the size of sample 2, x1 is the mean of sample 1, x2 is the mean of sample 2, d is the hypothesized difference between the population means, and SE is the standard error.
At 32 degree of freedom at 95% of confidence interval critical value of t is 1.694
Thus, the P-value < 0.005.
Interpret results. The mean for resting pluse rate for all women is greater than the mean resting pulse for men.
8.20 )
The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below:
State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: 1 - 2 = 0
Alternative hypothesis: 1 - 2 0
Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the difference between sample means is too big or if it is too small.
Formulate an analysis plan. For this analysis, the significance level is 0.10. Using sample data, we will conduct a two-sample t-test of the null hypothesis.
Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).
SE = sqrt[(s12/n1) + (s22/n2)]
SE = sqrt[((0.85)2/85) + ((0.75)2/52] = sqrt(0.0085 + 0.0108 ) = sqrt(0.0193) = 0.1389
DF = (s12/n1 + s22/n2)2 / { [ (s12 / n1)2 / (n1 - 1) ] + [ (s22 / n2)2 / (n2 - 1) ] }
DF = ((0.85)2/85 + (0.75)2/52)2 / { [ ((0.85)2 / 85)2 / (84) ] + [ ((0.75)2 / 52)2 / (51) ] }
DF = (0.0085 + 0.0108)2 / { [ (0.0085)2 / (84) ] + [ (0.0108)2 / (51) ] } = 0.00037 / (0.00000086 + 0.0000023) = 0.00037 / 0.00000315 = 117.46
t = [ (x1 - x2) - d ] / SE = [ (97.13 - 96.82) - 0 ] / 0.1389 = 0.31/0.1389 = 2.2318
where s1 is the standard deviation of sample 1, s2 is the standard deviation of sample 2, n1 is the size of sample 1, n2 is the size of sample 2, x1 is the mean of sample 1, x2 is the mean of sample 2, d is the hypothesized difference between the population means, and SE is the standard error.
t static value = 2.2318
t critical value = 2.402
The P-Value is .027525.
The result is not significant at p < .01
Interpret results. The claim is true the two means are different.
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.