Suppose that we wish to test the hypothesesHo : μ = 75H₁ : μ # 75for scores on a test using an a = 0.01 level of significance, when it is known that o = 5. Find the sample sizerequired if the power of our test is to be 0.90 when the true mean is 69.O 101915O 2011 A sample of the thickness (in hundreths of an inch) of 15 pieces of Fruit Flavored Bubble Gum yielded asample mean of 6.841 and a standard deviation of 0.155. A second independent sample of thethickness of 10 pieces of Fruit Flavored Gum yielded a sample mean of 6.701 and standard deviation of0.108. A researcher claims that Fruit Flavored Bubble Gum should be thicker on average than FruitFlavored Gum because bubble gum has more elasticity than regular gum. Assume both distributions arenormally distributed.Which test statistic is appropriate for testing the researcher's claim?O t':O=t =O ťt' =O t =O2 =(1-T₂)-do/s/n₁+82/12C-140S√n(T1-T2)-do/o/n₁+02/n2(1-2)-do8p√√/1/n₁+1/n₂x-Hoo/√n

Let x1 denote the sample of thickness for the Bubble Gum and x2 denote the sample of thickness for the Gum.The sample size, mean, and standard deviation for each sample are given as follwosx¯1=6.841s1=0.155n1=15x¯2=6.701s2=0.108n2=10To find:The appropriate test statistic for testing the claim that μ1>μ2The sample size for testing Ho:μ=75 and H1:μ≠75 whenα=0.01β=0.9μ=69σ=5.For the given study, the appropriate test statistic would be either a 2-sample t-test under the assumption of equal variance or a 2-sample t-test under the assumption of unequal variance. Consider the following R code for testing equality of variance using F test;1#Define the given standard deviations 2s1=0.155; s2=0.108 3n1=15; n2=10 4#Compute the F value 5F=s1^2/s2^2 6 7#compute the p-value 8p= pf(F, df1=15-1, df2=10-1, ncp=0, lower.tail=TRUE, log.p = FALSE) 9pvalue=2 * min(p, 1 - p) 10#take the decision at 5% level of significance 11if (pvalue < 0.05) { 12 cat("Reject Ho and conclude that variances are unequal") 13} else { 14 cat("Fail to reject Ho, hence the two variances are equal") 15}The R output is Hence the variances are assumed to be equal, then the appropriate test statistic is Explanation: Since there are 2 independent samples, whose variances are assumed to be equal, the t-test for pooled variance (sp2) would be the ideal choice for testing the researcher's claim.Incorrect options:Reason: Since there is not enough evidence to evidence to assume the unequal variance, this test statistic is not appropriate.Reason: This statistic is used when there is only one sample is included in the study design. Reason: The 2-sample t-test is used when the sample size is small and the population variances are unknown, but the above statistic includes the population variance which is not provided in the study.Reason: The test statistic is for one sample mean comparison when the population variance is provided. Since the population variance is not provided in the study the above test statistic is not appropriate for the given study.The formula to compute the minimum sample size based on the given formula is as followsHere the Z-critical values are computed using the Excel function =NORMSDIST(probability)For Z(1−α2)=Z(1−0.012), the Z-value is as followsFor Zβ=Z0.9, the Z-value is as followsThus,The required sample size is 11Test statistic which is appropriate for testing the researcher's claim is The required sample size is 11