Question

Construct the indicated confidence interval for the difference between the two population means. Assume that the two samples are independent simple random samples selected from normally distributed populations. Do not assume that the population standard deviations are equal.
A paint manufacturer wished to compare the drying times of two different types of paint. Independent simple random samples of 11 cans of type A and 9 cans of type B were selected and applied to similar surfaces. The drying times, in hours, were recorded. The summary statistics are as follows.
Type A
x-bar1 = 75.7 hrs.
s1 = 4.5 hrs.
n1 = 11
Type B
x-bar2 = 64.3 hrs.
s2 = 5.1 hrs.
n2 = 9
Construct a 98% confidence interval for the difference for the mean drying time between paint A and paint B.
A. 6.08 hrs < μ1 - μ2 < 16.72 hrs
B. 5.85 hrs < μ1 - μ2 < 16.95 hrs
C. 5.78 hrs < μ1 - μ2 < 17.02 hrs
D. 5.92 hrs < μ1 - μ2 < 16.88 hrs

Answer 1

Answer:

Step-by-step explanation:

The formula for determining the confidence interval for the difference of two population means is expressed as

Confidence interval = (x1 - x2) ± z√(s²/n1 + s2²/n2)

Where

x1 = sample mean of type A paint

x2 = sample mean of type B paint

s1 = sample standard deviation type A paint

s2 = sample standard for type B paint

n1 = number of samples of type A paint

n2 = number of samples of type B paint

From the information given,

x1 = 75.7

s1 = 4.5

n1 = 11

x2 = 64.3

s2 = 5.1

n2 = 9

x1 - x2 = 75.7 - 64.3 = 11.4

√(s1²/n1 + s2²/n2) = √(4.5²/11 + 5.1²/9) = √4.709

Degree of freedom = (n1 - 1) + (n2 - 1)

df = (11 - 1) + (9 - 1) = 18

For the 98% confidence interval, the z score from the t distribution table is 2.552

Margin of error = 2.552√4.709 = 5.55

The upper boundary for the confidence interval is

11.4 + 5.55 = 16.95 hours

The lower boundary for the confidence interval is

11.4 - 5.55 = 5.85 hours

The correct option is

B. 5.85 hrs < μ1 - μ2 < 16.95 hrs