Question

Suppose that we want to estimate the mean daily profit of a convenience store. For this purpose, we record the daily profit for a random sample of days. The sample has a mean of

468 dollars and a standard deviation of 76 dollars. For each of the following sampling scenarios, determine which test statistic is appropriate to use when making inference statements about the population mean.

(In the table, Z refers to a variable having a standard normal distribution, and t refers to a variable having a t distribution.)

Sampling Scenario	Z	t	could use either Z or t	unclear
(1) The sample has size 16, and it is from a population with a distribution about which we know very little.
(2) The sample has size 90, and it is from a non-normally distributed population with a known standard deviation of 75.
(3) The sample has size 85, and it is from a non-normally distributed population.
(4) The sample has size 10, and it is from a normally distributed population with unknown standard deviation.
(5) The sample has size 17, and it is from a normally distributed population with a known standard deviation of 75.

Answer 1

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1/
Population deviation is not know and sample size is samller than 30. Use t-distribution

2/
Population standard deviation is known. Use Z-distribution ( irrespective of sample size)

3/
sample size is large i.e. n > 30. Also, although the population is non-normal, according to CLT if sample size is large then normality can be assumed. Though population deviation isn't known, we can use either t or Z distribution

Use either Z or t

4/

small sample size of 10 and population is normal, which makes the sample also normal although the sample size is small. Use Z

5/
sample size is small, population is normal, deviation is known. Use Z distribution