Question

1. A random sample of 100 construction workers found that the average monthly salary was $21,310 USD with a standard deviation of $2405. A random sample of 81 Office workers found that the average salary was $27,041 with a standard deviation of $4651.

a.) Calculate a 95% confidence interval for the true mean difference between office workers’ salaries and construction workers’ salaries.

b.) Verify your conditions for inference

c.) Interpret your confidence interval with a sentence in context.

d.) According to your confidence interval, is there significant evidence of a difference between construction workers and office workers salaries?

2. A random sample of 100 patients with virus found that it was fatal in 18 people and 182 people eventually made a full recovery. In a random sample of 300 SARS patients, 34 died.  

a.) Verify your conditions for inference

b.) Calculate a 90% confidence interval for the true mean difference in the proportion of virus cases that were fatal and the proportion of SARS cases that were fatal.

c.) If you were to create a 95% confidence interval, would it be wider or narrower?

Answer 1

1. Let x1 be the random sample of office workers workers and x2 be the random sample of construction workers.

So we have : n1 = 81, = 27041, s₁ = 4651, n2 = 100, = 21310, s₂ = 2405

a) The 95% confidence interval for the true mean difference between office workers’ salaries and construction workers’ salaries is given by,

( - ) E

Where,

Here sample size is greater than 30, so we use normal distribution.

Here c = confidence level = 0.95, = 1-c = 1 -0.95 = 0.05

Zc = -----( using excel formula " =norm.s.inv(0.975 )" )

Hence margin of error is,

= 1117.20

Hence the 95% confidence interval is,

{ ( 27041-21310 ) - 1117.20 , ( 27041 -21310) + 1117.20 }

( 4613.8, 6848.2 )

b)

1. Here all samples are random samples.

2. Samples are independent.

3. Sampling distribution is approximately normal.

c) We have 95% confidence interval as ( 4613.8, 6848.2 ) .

Interpretation : We are 95% confident that difference between office workers’ salaries and construction workers’ salaries lies between 4613.8 and 6848.2.

d) Here we need to check, if there significant evidence of a difference between construction workers and office workers salaries.

The hypothesis are :

H0 : = or - = 0 v/s H1:    or   - 0

So here 0 does not lie the confidence interval ( 4613.8, 6848.2 ) .

Hence we say the H0 is rejected. Hence we say that there is significant difference between construction workers and office workers salaries.