Question

11. (10 pts total) One measure of quality and customer satisfaction is repeat business. A supplier of paper used for computer printouts obtained a random sample of 80 customer accounts from 2014 and found that 40 of these had placed more than one order during 2014. A similar survey conducted at the end of the 2015 revealed that 40 of 58 customers ordered again during 2015. (a) (8 pts) Do these data support the contention that there has been an increase in the proportion of repeat business from 2014 to 2015? Do a statistical test to see whether this contention is true (state the null and alternative hypotheses). Make your decision based on a level of significance of α = 0.05. Make sure you state the value of df if it is relevant. (b) (2 pts) Based on your answer to part (a), to what type of error are you now subject to? (Type I or Type II?)

Answer 1

(a) Hypothesis Test

Let p₁ = The Proportion of customers who placed more than one order in 2014. = 40 / 80 = 0.5

Let p₂ = The Proportion of customers who placed more than one order in 2015. = 40 / 58 = 0.86

Let = Overall proportion = (40 + 40)/(80 + 58) = 80 / 138 = 0.65

1 - = 0.35

= 0.05

(a) The Hypothesis:

H0: p₁ = p₂ : The proportion of customers who placed more than one order in 2014 is equal to the proportion of customers who placed more than one order in 2015.

Ha: p₁ < p₂ : The proportion of customers who placed more than one order in 2014 is lesser than the proportion of customers who placed more than one order in 2015.

This is a Left tailed Test.

The Test Statistic:

The p Value: The p value (Left tail) for Z = -4.38, is; p value = 0.0000

The Critical Value:   The critical value (Left tail) at = 0.05, Zcritical = -1.645

The Decision Rule: If Zobserved is <- Zcritical Then Reject H0.

Also If the P value is < , Then Reject H0

The Decision: Since Z observed (-4.38) is < Zcritical (1.645), We Reject H0.

Also since P value (0.0000) is < (0.05), We Reject H0.

The Conclusion: There is sufficient evidence at the 95% significance level to conclude that the proportion of customers who placed more than one order in 2014 is lesser than the proportion of customers who placed more than one order in 2015.

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(b) Since we have rejected the H0, it could be possible that we have wrongly rejected it.

The incorrect rejection of a true null hypothesis is a Type I error, which can be the error committed in this case.

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