Question

Two chemists working for a chicken fast-food company, have been producing a very popular sauce. Let’s call then Jesse and Mr. White. Gus, their boss, is tired of Mr. White’s negative attitude and is thinking about "firing" him and keeping only Jesse on the payroll. The problem, however, is that Mr. White seems to produce a higher quality sauce whenever he is in charge of production if compared to Jesse. Before making a final decision, Gus collected some data measuring the quality of different batches of sauce produced by Mr. White and Jesse. We assume the quality measurements for both of them are normally distributed with the common variance. The results, measured on a quality scale, are listed below:

Mr. White 97 1 7
Jesse 94 3 10
a. Based on this data, can we tell for sure (with 95% confidence) which one is the better chemist?
b. Gus wants to keep the mean quality score for the sauce above 90. In this case, can he can rid of Mr. White, i.e., is Jesse good enough to run the sauce production?

Answer 1

Check the explanation

Explanation:

Let
`\overline{x}` and
`\overline{y}` be sample means of white and Jesse denotes are two random variables.

Given that both samples are having normally distributed.

Assume
`\overline{x}` having with mean
`\mu_(1)` and
`\overline{y}` having mean
`\mu_(2)`

Also we have given the variance is constant

A)

We can test hypothesis as

`H0: \mu_(1) = \mu_(2)H1: \mu_(1) > \mu_(2)`

For this problem

Test statistic is

`T=\frac{\overline {x}-\overline {y}}{s\sqrt{\frac {1}{n1} +(1)/(n2)}}`

Where

`s=\sqrt{((n1-1)*s1^(2)+(n2-1)*s2^(2))/(n1+n2-2)}`

We have given all information for samples

By calculations we get

s=2.41

T=2.52

Here test statistic is having t-distribution with df=(10+7-2)=15

So p-value is P(t15>2.52)=0.012

Here significance level is 0.05

Since p-value is <0.05 we are rejecting null hypothesis at 95% confidence.

We can conclude that White has significant higher mean than Jesse. This claim we can made at 95% confidence.