Question

The mean I.Q of a sample of 1600 children was 99. Is it likely that this was a random sample from a population with a mean I.Q 100 and standard deviation of 15? (Test at 5% level of significance)

Answer 1

Sample size n = 1600

`bar(x)` = 99

Sample mean

Population mean µ = 100

Population S.D σ = 15

Under the Null hypothesis H₀: µ = 100

Alternative hypothesis H₁: µ = 100  ......(two tails)

Level of significance µ = 0.05

Test statistic z =`(bar(x) - mu)/sigma` N ∼ (0.1)

= `sigma/sqrt("n")`

z = `(99 - 100)/((15/sqrt(1600))`

= `(-1)/((15/40))`

= `(-1)/0.375`

z = – 2.666

z = – 2.67

Calculated value |z| = 2.67

Critical value at 5% level of significance is

`"Z"_(alpha/2)`  = 1.96

Inference: Since the calculated value is greater than table value i.e z

⇒ `"Z"_(alpha/2)` at 5% level of significance, the null hypothesis is rejected.

Therefore we conclude that the sample mean differs, significantly from the population mean.