Question

An overview of 30 crisis room patients found that the average waiting time for an ideal opportunity for treatment was 174.3 minutes. Expecting that the populace standard deviation is 46.5 minutes, find the best point estimate of the populace mean and the 99% confidence of the population mean.

Answer 1

99% confidence interval for the population mean is [152.39 min, 196.20 min].

Step-by-step explanation:

We are given that an overview of 30 crisis room patients found that the average waiting time for an ideal opportunity for treatment was 174.3 minutes.

Expecting that the population standard deviation is 46.5 minutes.

Firstly, the Pivotal quantity for a 99% confidence interval for the population mean is given by;

                                P.Q. =  $\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }$  ~ N(0,1)

where, $\bar X$ = sample mean waiting time for an ideal opportunity for treatment = 174.3 minutes

            $\sigma$ = population standard deviation = 46.5 minutes

            n = sample of crisis room patients = 30

           $\mu$ = population mean

Here for constructing a 99% confidence interval we have used One-sample z-test statistics as we know about population standard deviation.

So, 99% confidence interval for the population mean, $\mu$ is ;

P(-2.58 < N(0,1) < 2.58) = 0.99  {As the critical value of z at 0.5% level of

                                                    significance are -2.58 & 2.58}  

P(-2.58 < $\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }$ < 2.58) = 0.99

P( $-2.58 \times {\frac{\sigma}{\sqrt{n} } }$ < ${\bar X-\mu}$ < $2.58 \times {\frac{\sigma}{\sqrt{n} } }$ ) = 0.99

P( $\bar X-2.58 \times {\frac{\sigma}{\sqrt{n} } }$ < $\mu$ < $\bar X+2.58 \times {\frac{\sigma}{\sqrt{n} } }$ ) = 0.99

99% confidence interval for $\mu$ = [ $\bar X-2.58 \times {\frac{\sigma}{\sqrt{n} } }$ , $\bar X+2.58 \times {\frac{\sigma}{\sqrt{n} } }$ ]

                                         = [ $174.3-2.58 \times {\frac{46.5}{\sqrt{30} } }$ , $174.3+2.58 \times {\frac{46.5}{\sqrt{30} } }$ ]

                                         = [152.39 min, 196.20 min]

Therefore, 99% confidence interval for the population mean is [152.39 min, 196.20 min].