Question

The air in poultry processing plants has fungus spores. If there is not enough ventilation, this can affect the health of workers. The problem is most serious in the summer least serious in the winter. Here are data from the “kill room” of a plant that processes 37,000 turkeys a day taken on 4 separate days in the summer and the winter.

Summer 3175 2526 1763 1090

Winter 384 104 251 97
(a) Give a 99% confidence interval to estimate how much higher the mean count is during summer.
(b) Is there good evidence that the summer counts are higher than the winter counts?

Answer 1

(a) Give a 99% confidence interval to estimate how much higher the mean count is during summer

Solution:

Here, we have to find 99% confidence interval for the difference between the average counts for the summer and winter.

Confidence interval for difference between two population means is given as below:

Confidence interval = (X1bar – X2bar) ± t*sqrt[S_p²*((1/n1)+(1/n2))]

Where S_p² is pooled variance

S_p² = [(n1 – 1)*S1^2 + (n2 – 1)*S2^2]/(n1 + n2 – 2)

From given data, we have

X1bar = 2138.5

S1 = 906.4290743

X2bar = 209

S2 = 136.5747658

n1 = 4

n2 = 4

df = n1 + n2 – 2 = 4 + 4 – 2 = 6

Confidence level = 99%

Critical t value = 3.7074

(by using t-table)

S_p² = [(n1 – 1)*S1^2 + (n2 – 1)*S2^2]/(n1 + n2 – 2)

S_p² = [(4 – 1)* 906.4290743^2 + (4 – 1)* 136.5747658^2]/(4 + 4 – 2)

S_p² = 420133.1667

Confidence interval = (X1bar – X2bar) ± t*sqrt[S_p²*((1/n1)+(1/n2))]

Confidence interval = (2138.5 – 209) ± 3.7074*sqrt[420133.1667*((1/4)+(1/4))]

Confidence interval = 1929.5 ± 3.7074*458.3302

Confidence interval = 1929.5 ± 1699.2263

Lower limit = 1929.5 - 1699.2263 = 230.2737

Upper limit = 1929.5 + 1699.2263 = 3628.7263

Confidence interval = (230.2737, 3628.7263)

We are 99% confident that the difference between average count between summer and winter will lies between 230.2737 and 3628.7263.

(b) Is there good evidence that the summer counts are higher than the winter counts?

Yes, there is a good evidence that the summer counts are higher than the winter counts because the value of zero is not lies within the given confidence interval and all values are positive in nature.