Does Friday the $13th$have an effect on people's behavior? Researchers collected data on the numbers of shoppers at a sample of $45$different grocery stores on Friday the $6th$and Friday the $13th$in the same month. The dotplot and computer output below summarize the data on the difference in the number of shoppers at each store on these two days (subtracting in the order $6th$minus $13th$).

Researchers would like to carry out a test of $H_0:{\mu }_d=0$versus $H_{\alpha }:{\mu }_d\ne 0$, where ${\mu }_d$is the true mean difference in the number of grocery shoppers on these two days. Which of the following conditions for performing a paired $t$test is not met?

1. Random

II. Normal

III. Independent

(a) I only

(b) II only

(c) III only

d) I and II only

(e) I, II, and III

Step by step solution

01

Step-1 Given Information

Given in the question that,

the plot and output are:

we have to find the correct option.

02

Step-2 Explanation

Following are the main requirements for the paired t- test:

a. The samples taken from the population are simple random samples.

b. The size of the sample must be at least $30$.

c. Population must be at least $10\%$greater than the sample size.

In this case,

a. Data is gathered only on Friday and it is not possible that everyone goes out for the shopping on Friday. Thus, the sample is not random.

b. The sample size is $45$which is above $30$. Thus, normality condition is fulfilled.

c. Sample size of $45$is lower than the $10\%$of the size of the population.

Thus, the correct option is (a).

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