Question

Calculate a 95% confidence interval around your sample mean 29.7 24.5 27.1 29.8 29.2 27.0 27.8 24.1 29.3 25.9 26.2 24.5 32.8 26.8 27.8 24.0 23.6 29.2 26.5 27.7 27.1 23.7 24.1 27.2 25.9 26.7 27.8 27.3 27.6 22.8 25.3 26.6 26.4 27.1 26.1

Answer 1

Based on the sample data, the 95% confidence interval around the sample mean is approximately (26.07, 27.45).

The Confidence Interval shows the probability that a population parameter lies between two interval values.

Confidence Interval Formula:

`[ \bar{x} \pm Z \left( (s)/(√(n)) \right) ]`

Where:

`(\bar{x})` = the sample mean

`(Z)` = the critical value for a 95% confidence interval (which is 1.96)

`(s)` = the sample standard deviation

`(n)` = the sample size

We need the sample mean and standard deviation to calculate the confidence interval.

Sample Mean
`((\bar{x})): [ \bar{x} = (1)/(N) \sum_(i=1)^(N) x_i ]\[ \bar{x} = (29.7 + 24.5 + \ldots + 26.1)/(35) ] [ \bar{x} = 26.76 ]`

Sample Standard Deviation
`((s)): [ s = \sqrt{(1)/(N-1) \sum_(i=1)^(N) (x_i - \bar{x})^2} ]`

`[ s = \sqrt{((29.7-26.76)^2 + (24.5-26.76)^2 + \ldots + (26.1-26.76)^2)/(34)} ]`

`[ s = 2.036 ]`

Confidence interval:
`[ 26.76 \pm 1.96 \left( (2.036)/(√(35)) \right) ] [ 26.76 \pm 0.689 ]`

Thus, the 95% confidence interval around the sample mean is approximately (26.07, 27.45).