please can anyone help with question 3 Question 3. [10 Marks] An electrical firm

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please can anyone help with question 3

Question 3. [10 Marks] An electrical firm manufactures light bulbs of which the lifetime is ap- proximately normally distributed with a standard deviation of 40 hours. a) [4 marks) If a sample of 36 bulbs has an average lifetime of 780 hours and standard deviation of 44 hours, find a 93% confidence interval for the population mean lifetime of all bulbs produced by this firm. b) [4 marks] What is the sarnple size required if we want the width of the 93% CI for the population mean lifeime of all bulbs produced by this firm to be at most 16 hours? c) [2 marks] Use the results from part (b) to determine the sample size required if we want the width of the 93% CI for the population mean lifetime of all bulbs produced by this firm to be at most 8 hours (a half of 16 hours)?

Explanation / Answer

a)

CI for 93%

n = 36

mean = 780

z-value of 93% CI = 1.8119

std. dev. = 44

SE = std.dev./sqrt(n) = 7.33333

ME = z*SE = 13.28734

Lower Limit = Mean - ME = 766.71266

Upper Limit = Mean + ME = 793.28734

93% CI (766.7127 , 793.2873 )

b)

ME = width/2 = 16/2 = 8

Given CI Level = 93%

Margin of Error(ME) = 8

std. dev. = 44

z-value of 93% CI = 1.8119

n = (z*sigma/ME)^2 = 99.31

As n has to be whole number, round it to next whole number.

n = 100

c)

Here ME = 8/2 = 4

Given CI Level 93%

Margin of Error(ME) = 4

std. dev. = 44

z-value of 93% CI = 1.8119

n = (z*sigma/ME)^2 = 397.25

n = 398

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