Question

Reserve Problems Chapter 8 Section 2 Problem 1 Consider results of 20 randomly chosen people who have run a marathon. Their times, in minutes, are as follows: 141, 144, 150, 161, 169, 179, 186, 194, 199, 209, 219, 220, 226, 237, 254, 261, 275, 278, 286, 295. Calculate a 99% upper confidence bound on the mean time of the race. Assume distribution to be normal. Round your answer to the nearest integer (e.g. 9876).

Answer 1

99% Confidence interval: (183,246)

Explanation:

We are given the following data set:

141, 144, 150, 161, 169, 179, 186, 194, 199, 209, 219, 220, 226, 237, 254, 261, 275, 278, 286, 295

Formula:

`\text{Standard Deviation} = \sqrt{\displaystyle\frac{\sum (x_i -\bar{x})^2}{n-1}}`

where
`x_i` are data points,
`\bar{x}` is the mean and n is the number of observations.

`Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}`

`Mean =\displaystyle(4283)/(20) = 214.15`

Sum of squares of differences = 45922.55

`S.D = \sqrt{(45922.55)/(19)} = 49.16`

99% Confidence interval:

`\bar{x} \pm t_(critical)\displaystyle(s)/(√(n))`

Putting the values, we get,

`t_(critical)\text{ at degree of freedom 19 and}~\alpha_(0.01) = \pm 2.86`

`214.15 \pm 2.86((49.16)/(√(20)) ) = 214.15 \pm 31.44 =(182.71,245.59) \approx (183,246)`