Question

A sample of 100 students is drawn from a school. The mean weight and variance of the sample are 67.45 kg and 9 kg. respectively. Find (a) 95% (b) 99% confidence intervals for estimating the mean weight of the students

Answer 1

Sample size n = 100

The sample mean = `bar(x)` = 67.45

The sample variance S² = 9

The sample standard deviation S = 3

S.E = `"S"/sqrt("n")`

= `3/sqrt(100)`

= `3/10`

= 0.3

(a) The 95% confidence limits for µ are given by

`bar(x) - "Z"_(alpha/2)` S.E < µ < `bar(x) + "Z"_(alpha/2)` S.E

67.45 – (1.96 × 0.3) ≤ µ ≤ 67.45 + (1.96 × 0.3) 67.45 – 0.588 ≤ µ ≤ 67.45 + 0.588

66.862 ≤ µ ≤ 68.038

The confidential limits is (66.86, 68.04)

(b) The 99% confidence limits for estimating µ are given by

`bar(x) - "Z"_(alpha/2)` S.E ≤ µ ≤ `bar(x) + "Z"_(alpha/2)` S.E

67.45 – (2.58 × 0.3) ≤ µ ≤ 67.45 + (2.58 × 0.3)

67.45 – 0.774 ≤ µ ≤ 67.45 + 0.774

66.676 ≤ µ ≤ 68.224

∴ The 99% confidence limits is (66.68, 68.22)