You are the owner of two gym facilities in the area on A Road and B Road. Each l

Question

You are the owner of two gym facilities in the area on A Road and B Road. Each location has a lap pool with many regular swimmers. You have enough money in your budget to expand the lap pool at one of your locations this year. You decide to analyze the pool usage for a 10 day period and expand the pool that has the most swimmers. Assume the population standard deviations are the same. At the .05 significance level, is there a difference in the number of swimmers between your two locations?

a. State the null and alternate hypotheses.

b. Select alpha.

c. Select the test statistic.

d. Formulate the decision rule.

e. What is the value of the test statistic?

f. Determine and interpret the effect size and p-value.

g. Draw conclusions based on statistical and practical significance.

36 is included in group A

45 is included in group B

Road A Road B 47 56 42 57 32 49 41 41 39 62 28 35 24 31 39 38 45 49

36 is included in group A

45 is included in group B

Explanation / Answer

a. State the null and alternate hypotheses.

H0 : Number of swimmers at site A equals to number of swimmers at site B. A = B

Ha : Number of swimmers at site A is not equals to number of swimmers at site B. A B

b. Select alpha.

Alpha = 0.05

c. Select the test statistic.: Here we will use t test for equal variance.

Here for calculating test statistic.

First we have to calculate mean population at Road A and Road B and their standard deviation

Pooled Variance sp2 = (s1 2 + s22 )/2 = (7.362 + 10.1662 )/2 = 78.789

Pooled standard deviation sp = 8.87

Test Statistic

t = (x2 -x1 )/ (sp * sqrt (1/n1 + 1/n2) ]

t = (46.3 - 37.3)/ (8.87 * sqrt (1/10 + 1/10)) = 9/ 3.97 = 2.267

d. Formulate the decision rule.

We will reject the null hypothesis when t > tcritical

where tcritical = t18,0.05 = 2.101

e. What is the value of the test statistic?

Here t = 2.267

f. Determine and interpret the effect size and p-value.

Effect Size =  (M2 - M1) SDpooled = (46.3 - 37.3)/ 8.87 = 1.014

so effect size is pretty large here

p - value = 0.036 < 0.05 so we can reject the nul hypothesis .

g. Draw conclusions based on statistical and practical significance.

Here, we can reject the null hypothesis and conclude that there is population at Road A is significantly different than Road B.

Road A Road B 47 56 42 57 32 49 41 41 39 62 28 35 24 31 39 38 45 49 36 45 Mean 37.3 46.3 Std. Dev. 7.36 10.166

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