3 In 1999, a sample of 200 in-store shoppers showed that 42 paid by debit card. In 2004, a sample
of the same size showed that 62 paid by debit card. (a) Formulate appropriate hypotheses to test
whether the percentage of debit card shoppers increased. (b) Carry out the test at á = .01. (c) Find
the p-value. (d) Test whether normality may be assumed.
This problem fits a binomial proportion 2-sample z-test using proportions.
Hypotheses:
Ho: p1 = p2
Ha: p1 < p2 -->one-tailed test (shows a specific direction)
The formula is:
z = (p1 - p2)/√[pq(1/n1 + 1/n2)]
...where 'n' = sample sizes, 'p' is (x1 + x2)/(n1 + n2), and 'q' is 1-p.
I'll get you started:
p = (42 + 62)/(200 + 200) = ? -->once you have the fraction, convert to a decimal (decimals are easier to use in the formula)
q = 1 - p
p1 = 42/200
p2 = 62/200
Convert all fractions to decimals. Plug those decimal values into the formula and find z. Once you have this value, you will be able to determine the p-value or the actual level of this test statistic by using a z-table. Determine the outcome of the test (whether or not to reject the null). Draw your conclusions from there.
I hope this will help get you started.
(a) The null hypothesis (H0) can be formulated as: The percentage of debit card shoppers in 1999 and 2004 is the same. The alternative hypothesis (H1) can be formulated as: The percentage of debit card shoppers in 2004 is higher than in 1999.
(b) To carry out the test at α = .01, we need to perform a two-sample proportion test.
The formula for the test statistic is:
Z = (p̂2 - p̂1) / √(p̂(1 - p̂)((1/n1) + (1/n2)))
Where:
p̂1 = proportion of debit card shoppers in 1999 = 42/200 = 0.21
p̂2 = proportion of debit card shoppers in 2004 = 62/200 = 0.31
p̂ = pooled proportion = (x1 + x2) / (n1 + n2) = (42 + 62) / (200 + 200) = 0.265
n1 = sample size in 1999 = 200
n2 = sample size in 2004 = 200
Plugging these values into the formula, we can calculate the test statistic Z.
Z = (0.31 - 0.21) / √(0.265 * (1 - 0.265) * ((1/200) + (1/200)))
(c) To find the p-value, we need to compare the test statistic Z to the standard normal distribution. In this case, since we are testing if the percentage of debit card shoppers increased, we are looking for the p-value in the right tail of the distribution. Using a table or statistical software, we can find the p-value associated with the calculated Z score.
(d) To test whether normality may be assumed, we can use the Central Limit Theorem. Since the sample sizes in both years are large (n1 = 200 and n2 = 200), we can assume that the sampling distribution of the proportion difference is approximately normal.