STATS List below are the ages (years) of randomly selected race car drivers. Construct a 95% confidence interval estimate of the mean age of all race car drivers: ages 32, 40, 27, 36, 29, 28

Answer:

A 95% confidence interval estimate of the mean age of all race car drivers is [26.65 years, 37.35 years].

Step-by-step explanation:

We are given below the ages (in years) of randomly selected race car drivers;

Ages: 32, 40, 27, 36, 29, 28.

Firstly, the pivotal quantity for finding the confidence interval for the population mean is given by;

                           P.Q.  =  \(\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }\)  ~ \(t_n_-_1\)

where, \(\bar X\) = sample mean age = \(\frac{\sum X}{n}\) = \(\frac{32+40+27+36+29+28}{6}\) = 32 years

           s = sample standard deviation = \(\sqrt{\frac{\sum (X-\bar X)^{2} }{n-1} }\)  = 5.1 years

           n = sample of car drivers = 6

           \(\mu\) = population mean age of all race car drivers

Here for constructing a 95% confidence interval we have used a One-sample t-test statistics because we don't know about population standard deviation.

So, 95% confidence interval for the population mean, \(\mu\) is ;

P(-2.571 < \(t_5\) < 2.571) = 0.95  {As the critical value of t at 5 degrees of

                                                freedom are -2.571 & 2.571 with P = 2.5%}  

P(-2.571 < \(\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }\) < 2.571) = 0.95

P( \(-2.571 \times {\frac{s}{\sqrt{n} } }\) < \({\bar X-\mu}\) < \(2.571 \times {\frac{s}{\sqrt{n} } }\) ) = 0.95

P( \(\bar X-2.571 \times {\frac{s}{\sqrt{n} } }\) < \(\mu\) < \(\bar X+2.571 \times {\frac{s}{\sqrt{n} } }\) ) = 0.95

95% confidence interval for \(\mu\) = [ \(\bar X-2.571 \times {\frac{s}{\sqrt{n} } }\) , \(\bar X+2.571 \times {\frac{s}{\sqrt{n} } }\) ]

                                                  = [ \(32-2.571 \times {\frac{5.1}{\sqrt{6} } }\) , \(32+2.571 \times {\frac{5.1}{\sqrt{6} } }\) ]

                                                  = [26.65, 37.35]

Therefore, a 95% confidence interval estimate of the mean age of all race car drivers is [26.65 years, 37.35 years].

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