In 1999, a sample of 400 in store shoppers showed that 184 paid by debit card. I
ID: 3041429 • Letter: I
Question
In 1999, a sample of 400 in store shoppers showed that 184 paid by debit card. In 2013, a sample of the same size showed that 300 paid by debit card. Formulate the appropriate hypotheses to test whether the percentage of the debit card shoppers has increased. Carry out the test at the 0.01 significance level.
The problem involves:
Comparing the proportions of two populations
Comparing the variances of two populations
Comparing means of two independent populations
Comparing the means of two dependent samples
None of the above
What is the correct hypothesis setup for this problem?
a. H0: m2013 £ m1999 H1: m2013 > m1999
b. H0: m2013 ³ m1999 H1: m2013 < m1999
c. H0: p2013 £ p1999 H1: p2013 > p1999
d. H0: p2013 ³ p1999 H1: p2013 < p1999
e. None of the above
What is the sample proportion for 2013? ________ (round to 4 decimals)
What is the sample proportion for 1999? ________ (round to 4 decimals)
What is the combined proportion of success (who used debit card) for 1999 and 2013? _____ (round to 4 decimals)
Note: for questions #18 and #19, enter the data into the template in the order in which the years are presented on the hypotheses – that is, 2013 is first, then 1999 is second.
The computed test statistic for the sample above is ________. (round to 4 decimals).
At the 0.01 significance level, what is the critical value for testing the null hypothesis? _________ (round to 4 decimals)
What is the p-value for the test? _____________ (round to 4 decimals)
At the 0.01 significance level, what is the correct statistical conclusion?
Reject the null hypothesis
Fail to reject the null hypothesis
Reject the alternate hypothesis
Fail to reject the alternate hypothesis
None of the above. The test is inconclusive
Based on the result of the statistical test above, which of the following would be the correct conclusion?
The percentage of debit card users has increased from 1999 to 2013
The percentage of debit card users has decreased from 1999 to 2013
The percentage of debit card users has not changed from 1999 to 2013
Explanation / Answer
Solution:-
The problem involves: Comparing the proportions of two populations
P1999= 184/400 = 0.46
P2003= 300/400 = 0.75
a) State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: P1999> P2003
Alternative hypothesis: P1999 < P2003
Note that these hypotheses constitute a one-tailed test.
Formulate an analysis plan. For this analysis, the significance level is 0.01. The test method is a two-proportion z-test.
Analyze sample data. Using sample data, we calculate the pooled sample proportion (p) and the standard error (SE). Using those measures, we compute the z-score test statistic (z).
p = (p1 * n1 + p2 * n2) / (n1 + n2)
p = 0.605
SE = sqrt{ p * ( 1 - p ) * [ (1/n1) + (1/n2) ] }
SE = 0.034567
z = (p1 - p2) / SE
z = - 8.39
where p1 is the sample proportion in sample 1, where p2 is the sample proportion in sample 2, n1 is the size of sample 1, and n2 is the size of sample 2.
Since we have a one-tailed test, the P-value is the probability that the z-score is less than -8.39.
Thus, the P-value = less than 0.0001
Interpret results. Since the P-value (almost 0) is less than the significance level (0.01), we have to reject the null hypothesis.
Reject the null hypothesis
The percentage of debit card users has increased from 1999 to 2013.
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