Question

A random sample of 35 transmission replacement costs find the mean to be $2520.00.

Assume the population standard deviation is $450.00.

A 95​% confidence interval for the population mean repair cost is (2370.9,2669.1).

Change the sample size to n=70.

Construct a 95​% confidence interval for the population mean repair cost. Which confidence interval is​ wider? Explain.

Construct a 95​% confidence interval for the population mean repair cost.

The 95​% confidence interval is ​(...​,...​). ​(Round to one decimal place as​ needed.)

Answer 1

Solution

Back-up Theory

100(1 - α) % Confidence Interval for μ, when σ is known:

Xbar ± MoE……………………………………........................................................................…………………………….. (1)

Where

MoE = (Z_{α /2})σ/√n ………………………………....................................................................…………………………..... (2)

With

Xbar = sample mean,

Z_{α /2} = upper (α /2)% point of N(0, 1),

σ = population standard deviation and

n = sample size.

Now, to work out the solution,

There are two ways of getting the confidence interval with sample size n = 70.

One way is to use the formula (1) given above.

The second method, which is faster and more intellectual, would be use the underlying principle of CI.

Solution by the second method:

First we note that n affects the MoE only.

All other things remaining the same, if M₁ is the MoE with n = n₁ and M₂ is the MoE with n = n₂,

then, given M₁, M₂ = M₁ x √(n₁/n₂) ................................................................................................................................. (3)

Now, MoE = Upper bound of CI – Xbar or also = Xbar – lower bound of CI.

So, for n = 35, MoE. M₁ = 2669.1 – 2520 = 149.1

Hence, vide (3), for n = 70, MoE. M₂ = 149.1 x √(35/70)

= 149.1/√2

= 105.4296

Thus, 95​% confidence interval for the population mean repair cost for a sample of size 70 is: [2414.5704, 2625.4296]

i.e., [2414.6, 2625.4] Answer

DONE

Going beyond,

Details of calculations by Method 1

Given	α	0.05	1 - (α/2) =	0.975
	n	70	SQRT(n)	8.36660027
	σ	450
	Xbar	2520
	Zα/2	1.959963985
95% CI for μ		2520	±	105.417226
Lower Bound		2414.582774
Upper Bound		2625.417226

Complete