Question

A paint manufacturer wants to determine the average drying time of a new interior wall paint. For n = 50 test areas of equal size he obtained a sample mean drying time of ¯x = 63.3 minutes and a sample standard deviation of s = 8.4 minutes. (a) (5 points) Construct the 95% confidence interval for the true mean µ. (b) (5 points) Suppose the population has a normal distribution, construct the exactly 95% confidence interval for the true mean µ. (c) (5 points) Suppose the population has a normal distribution, construct the 95% prediction interval for a new observation.

Answer 1

To construct a 95% confidence interval for the true mean drying time of a new interior wall paint, use the formula: CI = ¯x ± z * (s / sqrt(n)). For an exact 95% confidence interval assuming a normal distribution, use the same formula. To construct a 95% prediction interval for a new observation, use the formula: PI = ¯x ± z * s.

Step-by-step explanation:

(a) To construct a 95% confidence interval for the true mean, we use the formula:

CI = ¯x ± z * (s / sqrt(n))

where ¯x is the sample mean, s is the sample standard deviation, n is the sample size, and z is the z-score for the desired confidence level.

Plugging in the values, we get:

CI = 63.3 ± z * (8.4 / sqrt(50))

Using a z-score table or calculator, we find that the z-score for a 95% confidence level is approximately 1.96.

So, the 95% confidence interval for the true mean drying time is:

(63.3 - 1.96 * (8.4 / sqrt(50)), 63.3 + 1.96 * (8.4 / sqrt(50)))

(b) Assuming the population has a normal distribution, we can use the same formula as in part (a) to construct an exact 95% confidence interval.

(c) To construct a 95% prediction interval for a new observation, we use the formula:

PI = ¯x ± z * s

Where ¯x and s are the same as in the previous parts, and z is the z-score for the desired confidence level.

Again, using a z-score table or calculator, we find that the z-score for a 95% confidence level is approximately 1.96.

So, the 95% prediction interval for a new observation is:

(63.3 - 1.96 * 8.4, 63.3 + 1.96 * 8.4)

Answer 2

a) the 95% confidence interval for the true mean µ is 60.971, 65.629

The exact 95% confidence interval for the true mean µ is 60.971, 65.629

P (x`- ∝ s/√n<u <x`+ ∝ s/√n) = 0.95

Step-by-step explanation:

n= 50

x`= 63.3

s= 8.4 minutes

∝= 95%= ±1.96

The formula for calculating the confidence interval is:

x`± ∝ s/√n

Putting the values

x`± 1.96 s/√n

63.3 ± 1.96(8.4/√50)

63.3± 2.3287

60.971, 65.629

a) the 95% confidence interval for the true mean µ is 60.971, 65.629

b) Putting the values

x`± 1.96 s/√n

63.3 ± 1.96(8.4/√50)

63.3± 2.3287

The exact 95% confidence interval for the true mean µ is 60.971, 65.629

c) The prediction level tells that the drying time of the wall from 60.971 to 65.629 minutes must be in the range 95 % of the time.

And is given by

P (x`- ∝ s/√n<u <x`+ ∝ s/√n) = 0.95