Question

A sample of 26 offshore oil workers took part in a simulated escape exercise, resulting in the accompanying data on time (sec) to complete the escape.
389 357 359 364 375 424 326 395 402 373
374 371 365 367 365 326 339 393 392 369
374 359 357 403 335 397

A normal probability plot of the n 26 observations on escape time given above shows a substantial linear pattern; the sample mean and sample standard deviation are 371.08 and 24.45, respectively. (Round your answers to two decimal places.)

Required:
a. Calculate an upper confidence bound for population mean escape time using a confidence level of 95%.
b. Calculate an upper prediction bound for the escape time of a single additional worker using a prediction level of 95%.

Answer 1

Answer:

The upper confidence bound for population mean escape time is: 379.27

The upper prediction bound for the escape time of a single additional worker  is 413.64

Step-by-step explanation:

Given that :

sample size n = 26

sample mean $\bar x$ =  371.08

standard deviation $\sigma$ = 24.45

The objective is to calculate an upper confidence bound for population mean escape time using a confidence level of 95%

We need to determine the standard error of these given data first;

So,

Standard Error S.E = $\dfrac{\sigma }{\sqrt{n}}$

Standard Error S.E = $\dfrac{24.45 }{\sqrt{26}}$

Standard Error S.E = $\dfrac{24.45 }{4.898979486}$

Standard Error S.E = 4.7950

However;

Degree of freedom df= n - 1

Degree of freedom df= 26 - 1

Degree of freedom df= 25

At confidence level of 95% and Degree of freedom df of  25 ;

t-value = 1.7080

Similarly;

The Margin of error = t-value × S.E

The Margin of error = 1.7080 × 4.7950

The Margin of error = 8.18986

The upper confidence bound for population mean escape time is = Sample Mean + Margin   of  Error

The upper confidence bound for population mean escape time is =  371.08 +  8.18986

The upper confidence bound for population mean escape time is = 379.26986  $\approx$ 379.27

The upper confidence bound for population mean escape time is: 379.27

b.  Calculate an upper prediction bound for the escape time of a single additional worker using a prediction level of 95%.

The standard error of the mean = $\sigma \times \sqrt{1+ \dfrac{1}{n}}$

The standard error of the mean = $24.45 \times \sqrt{1+ \dfrac{1}{26}}$

The standard error of the mean = $24.45 \times \sqrt{1+0.03846153846}$

The standard error of the mean = $24.45 \times \sqrt{1.03846153846}$

The standard error of the mean = $24.45 \times 1.019049331$

The standard error of the mean = 24.91575614

Recall that : At confidence level of 95% and Degree of freedom df of  25 ;

t-value = 1.7080

∴

The Margin of error = t-value × S.E

The Margin of error = 1.7080 × 24.91575614

The Margin of error = 42.55611149

The upper prediction bound for the escape time of a single additional worker  is calculate by the addition of

Sample Mean + Margin of Error

= 371.08 + 42.55611149

= 413.6361115

$\approx$ 413.64

The upper prediction bound for the escape time of a single additional worker  is 413.64