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# Test of hypotheses: difference between means and proportions

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1) SafeRoad Stores is redesigning the checkout lanes in its stores throughout the country. Two designs have been suggested. Tests on customer checkout times have been conducted at two stores where the two new systems have been installed. A summary of the sample data follows:
System A System B
Sample size = 25 Sample size = 25
Sample mean checking out time = 4.3 minutes Sample mean checkout time = 3.3 minutes
Sample standard deviation = 2.0 minutes Sample standard deviation = 3.0 minutes

At the 0.05 significance level, perform a statistical test to see the two systems are significantly different.

2) Suppose the Real Estate Board of Greater Vancouver conducts a study of the number of first-time applicants who are successful in passing the certification examination for real estate salespeople on their first attempt. Two samples were taken from the list of applicants: those who were already employed by a real estate agency and those who were recent university graduates who had not taken a full-time job. Results are shown in the table below:

Real Estate Employees Unemployed Graduates Sample Size Number passing
100 47 200 102

a. What is the sample proportion of real estate employees who passed the exam on their first attempt?

b. What is the sample proportion of unemployed college graduates who passed the exam on their first attempt?

c. If the null hypothesis is H0: proportion R - proportion U = 0, what is the value of the test statistic? proportionR = proportion of real estate employees passing the exam on their first attempt, proportion U = proportion of unemployed graduates passing the exam on their first attempt.

d. State your conclusion on the result of the test for H0: proportion R - proportion U = 0 if the alternate hypothesis is: H0: proportion R - proportion U < 0. Use alpha (a) = 0.05.

##### Solution Summary

The expert tests hypotheses for difference between means and proportions. The significance levels are performed.

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1. SafeRoad Stores is redesigning the checkout lanes in its stores throughout the country. Two designs have been suggested. Tests on customer checkout times have been conducted at two stores where the two new systems have been installed. A summary of the sample data follows:

System A System B
Sample size = 25 Sample size = 25
Sample mean checking out time = 4.3 minutes Sample mean checkout time = 3.3 minutes
Sample standard deviation = 2.0 minutes Sample standard deviation = 3.0 minutes

At the 0.05 significance level, perform a statistical test to see the two systems are significantly different.

Small Sample Size (Independent Sample)

Data
Sample 1: System A
Mean of sample 1= M 1 = 4.3 minutes
Standard deviation of sample 1= s1 = 2.0 minutes
Sample size of sample 1= n1 = 25

Sample 2: System B
Mean of sample 2= M 2 = 3.3 minutes
Standard deviation of sample 2= s2 = 3.0 minutes
Sample size of sample 2= n2 = 25
Hypothesized difference between means = 0
Significance level= 0.05

1) Hypothesis
Null Hypothesis: Ho: M 1 = M 2 (M 1 is equal to M 2)
Alternative Hypothesis: H1: M 1 not equal to M 2 :( M 1 is not equal to M 2 )
No of tails= 2 (Both tails )
This is a 2 tailed (Both tails ) test because we are testing that M 1 not equal to M 2
Significance level=alpha (a) = 0.05 or 5%

2) Decision rule
we use t distribution as we are dealing with small sample sizes
t at the 0.05 level of significance and 48 degrees of freedom (=n1+n2-2) and 2 tailed test= 2.0106
t critical = ± 2.0106

if sample statistic is <-2.0106 or > 2.0106, Reject Null Hypothesis, else Accept Null Hypothesis
Alternatively
if p value is less than the significance level (= 0.05 ), Reject Null Hypothesis, else Accept Null Hypothesis

3) Calculation of sample statistics
Sample 1: Sample 2:
System A System B
Mean =M = 4.30 3.30
Standard deviation =s= 2.00 3.00
Sample size=n= 25 25

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