Question

A sample of 400 individuals is found to have a mean height of 67.47 inches. Can it be reasonably regarded as a sample from a large population with a mean height of 67.39 inches and standard deviation of 1.30 inches?

Answer 1

Sample size n = 400; sample mean `bar(x)` = 67.47 inches

Sample SD S = 1.30 inches

Population mean µ = 67.39 inches

population SD σ = 1.30 inches

Null Hypothesis H₀: µ = 67.39 inches ......(the sample has been drawn from the population mean µ = 67.39 inches; population SD σ = 1.30 inches)

Alternative Hypothesis H₁ = µ ≠ 67.39 inches ......(two tail)

i.e The sample has not been drawn from the population mean µ = 67.39 inches and SD σ = 1.30 inches

The level of significance α = 5% = 0.05

Test static:

z = `(67.47 - 67.39)/((1.30/sqrt(400))`

= `(0.08/(1.30/20))`

= `0.08/0.065`

= 1.23076

= 1.2308

Thus the calculated and the significant value or `"Z"_(("a")/2)` = 1.96

Table value comparing the calculated and table values `"Z"_(("a")/2)`

i.e., 12308 < .96

Inference: Since the calculated value is less than value

i.e `"Z" > "Z"_(("a")/2)` at 5% level of significance, the null hypothesis is accepted Hence we conclude that the data doesn’t provide us any evidence against the null hypothesis.

Therefore, the sample has been drawn from the population mean µ = 67.39 inches and SD σ = 1.30 inches.