Question 1 (a)The summary statistics for the variable duration (Sec) is presented below: Confidenceinterval=Mean±Confidencelevel(93.0%) Upperlimit=229.61+4.08=233.69 Lowerlimit=229.61−4.08=225.53 Hence, 93% confidence interval is[225.53233.69] The t value with the help of T.INV.2T has computed through excel and is highlighted below: 1
Thet∝ 2value comes out to be 2.120. (b)The summary statistics is shown below: 97%Confidenceinterval=Mean±Confidencelevel(97.0%) Upperlimit=−3.28+0.82=−2.46 Lowerlimit=−3.28−0.82=−4.11 Hence, 93% confidence interval is[−2.46−4.11] Question 2 a)The requisite hypotheses are as stated below. Null Hypothesis (H0): μ ≥ 7 i.e. average rating of men would be atleast 7 out of 10 Alternative Hypothesis (H1): μ < 7 i.e. average rating of men would be lesser than 7 out of 10 2
b)The relevant test statistic for this test would be z as the population variance is known and the sample size is greater than 30. Relevant Formula for Z test statistic = (X-μ)/SE In the given case, X = 6.14, μ = 7, SE = σ/√n = 1.73/√187 = 0.1265 Hence, calculated Z statistic = (6.14 -7)/(0.1265) = -6.80 c)In accordance with the table A-2, the necessary p value for the above value of test statistic comes out as 0.0001. d)It is apparent that the computed p value (0.0001) is lower than the given significance level of 1% or 0.01. As a result, there exists enough statistical evidence to cause null hypothesis rejection and acceptance of alternative hypothesis. Thus, it would be appropriate to conclude that average attractiveness rating of men is lower than 7. e)It would be inappropriate to consider that 99% of the males would get an attractiveness rating lesser than 7. This is one of the shortcomings of the inferential statistical techniques as they tend to ignore the inherent difference in the variable under consideration and tend to provide an average view which may not apply to the concerned individual with a high consistency. This is especially the case for men attractiveness rating where there would be considerable difference amongst individuals. Question 3 a)The requisite hypotheses are indicated below. Null Hypothesis (H0) μ = 60 seconds 3
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Alternative Hypothesis (H1): μ ≠ 60 seconds b)The requisite values from excel are as follows c)The requisite computations manually are shown below. Sample Mean = 61.83 seconds, Sample standard deviation = 8.26 seconds Sample size = 6 Standard Error = Sample standard deviation/Sample size0.5= 8.26/(60.5) = 3.37 Hence, computed value of t statistic = (61.83-60)/3.37 = 0.543 The above is a two tail test and the p value comes out as 0.61 d)Based on the computation highlighted above, it is apparent that the p value is higher than the level of significance. This implies that the available evidence is not sufficient to cause 4
rejection of null hypothesis. Hence, the alternative hypothesis would not be accepted.. Thus, the original claim has been validated through hypothesis testing. Question 4 Number of men finished marathonn1= 24,521 Number of men dropped out marathonx1= 283 Number of women finished marathonn2= 12,853 Number of women dropped out marathonx2= 152 Significance level = 0.05 (a)Hypotheses Null hypothesisHo:p1−p2=0 Alternative hypothesisH1:p1−p2≠0 (b) Computation of test statistics and the critical value ^p1=x1 n1 =283 24521=0.0115 ^p2=x2 n2 =152 12853=0.0118 Pooled sample proportion for difference in the proportion is presented below: p=x1+x2 n1+n2 =283+152 24521+12853=0.011639 q=1−p=1−0.011639=0.988361 5
Pooled standard error is calculated below: SE=√pq n1 +pq n2 ¿√(0.011639∗0.988361) 24521+(0.011639∗0.988361) 12853 ¿0.0011679 Test statistics zobserved=^p1−^p2 SE=(0.0115−0.0118) 0.0011679=−0.24393 Therefore, the test statistics comes out to be−0.24393. The p value for upper tailed test pvalue=2∗P(Z<|z|)=2∗P(Z←0.24393)=0.8072 p value is higher than significance level and thus, null hypothesis would not be rejected. Critical value approach Critical valuezc=±z∝ 2 =±z0.05 2 =±1.96 (c) 95% confidence interval ^p1=x1 n1 =283 24521=0.0115 6
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^p2=x2 n2 =152 12853=0.0118 ^p1−^p2=0.0115−0.0118=−0.0003 SEp1−p2 =√^p1(1−^p1) n1 +^p2(1−^p2) n2 ¿√0.0115(1−0.0115) 24521+0.0118(1−0.0118) 12853 ¿0.001172 zc=1.96 Margin of error¿1.96∗0.001172=0.002297 Confidence interval¿(^p1−^p2)±(zc∗SEp1–p2) Upper limit¿−0.0003+(1.96∗0.001172)=0.0020 Lower limit¿−0.0003−(1.96∗0.001172)=−0.0026 Hence, 95% Confidence interval¿[−0.00260.0020] It can be concluded that confidence interval comprises 0 and hence, it can be said that “difference between the true proportions is zero and thus, no significant difference is present.” (d) It can be concluded from the above (hypothesis and confidence interval) that both men and women has finished the marathon at the same rate proportion. Question 5 The information and data can be summarized below: Sample 1 n1=14 7
x1=4 s1=0.829 Sample 2 n2=15 x2= 4.2 s2=0.328 (a)Hypotheses Null hypothesisHo:μ1=μ2 Alternative hypothesisH1:μ1>μ2 (b)Computation of test statistics and the critical value Pooled standard deviationsp=√(n1−1)s1 2+(n2−1)s2 2 n1+n2−2 sp=√(14−1)(0.829)2+(15−1)(0.328)2 14+15−2 sp=0.6218 The value oftest statisticst=x1−x2 sp√1 n1 +1 n2 8
t=4−4.2 0.6218√1 14+1 15 =−0.8655 Critical value for 5% significance level¿±1.7032 (c)It is apparent that for right tailed hypothesis the critical value is +1.7032 and thus, it can be seen that test statistics is lower than the critical value and thus, null hypothesis would not be rejected.Hence, the claim is incorrect that “the female and male professors have different mean evaluation rating.” (d)95% confidence interval Confidence interval¿(x1−x2)±(tc∗sx1−x2) sx1−x2 =√s1 2+s2 2 n1+n2−2=√(0.829)2+(0.328)2 14+15−2=0.17157 Upper limit¿−0.2+(1.7032∗0.17157)=0.0922 Lower limit¿−0.2−(1.7032∗0.17157)=−0.492 Hence, 95% Confidence interval¿[−0.4920.0922] It can be concluded that confidence interval comprises 0 and hence, it can be said that null hypothesis would not be rejected. Thus, the claim is incorrect that “the female and male professors have different mean evaluation rating.” (e)Using both the methods i.e. critical value and confidence interval, it is apparent that the available evidence does not warrant null hypothesis rejection. Hence, the claim made is with regards to difference in evaluation rating for male and female professors are incorrect. 9
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Question 6 Height of president and opponents are show below: (a)Significance level = 0.10 Hypotheses Null hypothesisHo:μHeightofpresident−μHeightofopponent=0 Alternative hypothesisH1:μHeightofpresident−μHeightofopponent>0 Value of t statistics Sum of differences∑d=13−2+0+15+8−6=28 Number of observationsn=6 d=∑d n=28 6=4.67 Standard deviationSd=√1 n−1∑¿¿¿ 10
Sd=√1 6−1∗367.33 Sd=8.57 Now, Test statisticst=d Sd √n t=4.67 8.57 √6 t=1.33 Degree of freedom¿n−1=6−1=5 The p value The p value with respect to t stat and degree of freedom and for on tailed hypothesis comes out to be 0.12002. Decision rule and conclusion It can be seen from the above that p value is higher than the significance level (0.12002 > 0.10) and thus, null hypothesis would not be rejected. Therefore, it can be concluded that the claim that difference in the mean heights of president is higher than the heights of opponents is incorrect. (b) Confidence interval Confidence interval¿d±tSd √n The value t for 95% confidence interval¿2.5706 11
Upperlimitof95%confidence=4.67+(2.5706)8.57 √6=13.66 Lowerlimitof95%confidence=4.67−(2.5706)8.57 √6=−4.32 Hence, the 95% confidence interval¿[−4.3213.66] It can be seen from the computed confidence interval that the confidence interval also contains zero and thus, null hypothesis would not be rejected. Therefore, it can be concluded that the claim that difference in the mean heights of president is higher than the heights of opponents is incorrect. Question 7 Given data Hypotheses Null hypothesisHo:“Different days of week have same frequencies of the police calls.” Alternative hypothesisH1:“Different days of week have different frequencies of the police calls.” Degree of freedom¿n−1=7−1=6 Level of significance =5% Chi-square test statistics would be used to test the hypothesis in present case. 12
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OiEiOi−Ei(Oi−Ei)2(Oi−Ei)2 Ei 208207.2860.7140.5100.002 224207.28616.714279.3671.348 246207.28638.7141498.7967.231 173207.286-34.2861175.5105.671 210207.2862.7147.3670.036 236207.28628.714824.5103.978 154207.286-53.2862839.36713.698 Sum31.963 Chi- square statistics ¿31.963 Critical value of chi-square for degree of freedom 6 and significance level 1% comes out to be 16.81. It can be seen that critical value is lower than calculated chi square statistics and thus, null hypothesis would be rejected and alternative hypothesis would be accepted. Therefore, the conclusion can be made that “Different days of week have different frequencies of the police calls.” 13