Question

A microbrewery makes a beer that contains an alcohol level between 4 and 6 degrees, assumed to be normally distributed. During the 20 fermentation days, the content of each tank is regularly brewed so that it remains homogenous. Next, the content of each tank is filtered and bottled. Ten bottles then are randomly selected. For a given sample, the level of alcohol in each bottle was as follows: 5 6 4 4 5 6 5.5 4 4.3 5.2 (a) For a significance level of 1%, can we conclude that the average level of alcohol in the bottles is different from 5 degrees? (b) For a significance level of 1%, can we conclude that the average level of alcohol in the bottles is less than 5 degrees? (c) Now, the accountant of the microbrewery proposes the following decision rule: The tank is unacceptable if X  5  z0.025 , and acceptable otherwise. 0.5 25 And the treasurer proposes the rule: 1 The tank is unacceptable if X  5  z0.025 , and acceptable otherwise. 0.05 25 (Notice that for any number a,|a|a. For instance,|5||5|5. z0.025 is a standard normal score.) What would be the decision under the rule (i) of the accountant? (ii) of the treasurer?

Answer 1

part 3 is not clear. I am answering the first 2 parts.

1) The null hypothesis is H₀ : average t = 5

H_A : average t is not equal to 5

This is a 2 tailed t test (as sample size is small)

For the 10 datapoints, the sample average is 4.9 and sample standard deviation is 0.794 (df =9 note)

The t statistic =Mod ( n^1/2 *(xbar - 5)/ Sample standard deiation) = 0.4

The tabulated value of t with 9 degrees of freedom and significance of 0.5% (two tailed) = 3.25

As calculated t is less than tabulated t it can be inferrred that the average temperature is not significantly different from 5 degrees

2) Here the difference is that the alternate hypothesis is average temperature is less than 5 (hence one sided t test)

The critical value now is t (9 degrees of freedom and level of significance of 1%) = 2.821

Again there is no information which rejects the null hypothesis. Hence the average temperature is not significantly less than 5 degrees