Question

(CO 4) In a sample of 8 high school students, they spent an average of 24.8 hours each week doing sports with a sample standard deviation of 3.2 hours. Find the 95% confidence interval, assuming the times are normally distributed.

(21.60, 28.00)

(22.12, 27.48)

(22.66, 26.94)

(24.10, 25.50)

Answer 1

(22.12, 27.48)

Explanation:

Given : Significance level :
`\alpha: 1-0.95=0.05`

Sample size : n= 8 , which is a small sample (n<30), so we use t-test.

Critical values using t-distribution:
`t_(n-1,\alpha/2)=t_(7,0.025)=2.365`

Sample mean :
`\overline{x}=24.8\text{ hours}`

Standard deviation :
`\sigma=3.2\text{ hours}`

The confidence interval for population means is given by :-

`\overline{x}\pm t_(n-1,\alpha/2)(\sigma)/(√(n))`

i.e.
`24.8\pm(2.365)(3.2)/(√(8))`

`24.8\pm2.67569206001\\\\\approx24.8\pm2.68\\\\=(24.8-2.68, 24.8+2.68)=(22.12, 27.48)`

Hence, the 95% confidence interval, assuming the times are normally distributed.= (22.12, 27.48)