Question

Beer bitterness (5 parts, 19 marks)

The taste of beers and its bitterness depends on the combination of its ingredients, mainly malts, hops, and yeast. The bitterness of beers is specified by the International Bitterness Unit (IBU) which measures the parts per million of a specific acid (isomerized alpha acid) found in one liter of beer. Lower IBU corresponds to less bitterness and higher IBU corresponds to more bitterness. In the Beer Bitterness dataset, you can find the level of bitterness measured in milligrams of isomerized alpha acid in 1 liter of beer for two samples from two types of beer (Midnight Ottawa and Midnight Montreal).

1. [2 Marks] Examine the boxplot of the level of bitterness for the sample of Midnight Ottawa and the boxplot of the level of bitterness for the sample of Midnight Montreal. Copy the boxplot and comment on the population distribution of the level of bitterness for each case.
2. [7 Marks] The manager of the brewery believes that Midnight Ottawa and Midnight Montreal differ in terms of bitterness. Carry out a hypothesis test to see whether there is a real difference in the mean level of bitterness of these two types of beer. You may use MS Excel or other software for your computations and copy the result. Assume the two population variances are unequal and use a 1% significance level.
3. [2 Marks] Find the corresponding 99% confidence interval for the difference in level of bitterness. Show your computations.
4. [1 Mark] Describe in one sentence how the hypothesis test in part b) and the confidence interval in part c) reflect the relationship between confidence intervals and hypothesis tests as introduced in class.
5. [7 Marks] The manager of the brewery thinks that assuming unequal population variances is wrong. Repeat part b), now manually, assuming equal population variances. Show your manual calculations and clearly state your conclusion. (Hint. You may use any statistics you need from part b)).

Beer Bitterness Data

	Level of IBU/ milligram of isomerized alpha acid in 1 liter of beer
Number of observation	Midnight Ottawa	Midnight Montreal
1	10.3	11.5
2	10.2	11.4
3	10.5	11.7
4	10.8	11.6
5	10.8	11.3
6	10.1	11.7
7	10.9	11.3
8	10.3	11.5
9	10.5	11.7
10	10.4	11.4
11	10.7	11.7
12	10.3	11.7
13	10.8	11.9
14	10.6	11.4
15	10.4	11.3
16	10.3	11.6
17	10.7	11.1
18	11	11.4
19	10.4	11.9
20	10.6
21	11
22	10.3
23	10.7
24	10.4

Answer 1

1. [2 Marks] Examine the boxplot of the level of bitterness for the sample of Midnight Ottawa and the boxplot of the level of bitterness for the sample of Midnight Montreal. Copy the boxplot and comment on the population distribution of the level of bitterness for each case.

The boxplots are:

The distribution has no outliers.

1. [7 Marks] The manager of the brewery believes that Midnight Ottawa and Midnight Montreal differ in terms of bitterness. Carry out a hypothesis test to see whether there is a real difference in the mean level of bitterness of these two types of beer. You may use MS Excel or other software for your computations and copy the result. Assume the two population variances are unequal and use a 1% significance level.

The hypothesis being tested is:

H₀: µ₁ = µ₂

H₁: µ₁ ≠ µ₂

Midnight Ottawa	Midnight Montreal
10.548	11.532	mean
0.261	0.216	std. dev.
23	19	n
	39	df
	-0.9838	difference (Midnight Ottawa - Midnight Montreal)
	0.0736	standard error of difference
	0	hypothesized difference
	-13.364	t
	3.85E-16	p-value (two-tailed)

Since the p-value (0.0000) is less than the significance level (0.01), we can reject the null hypothesis.

Therefore, we can conclude that there is a real difference in the mean level of bitterness of these two types of beer.

1. [2 Marks] Find the corresponding 99% confidence interval for the difference in level of bitterness. Show your computations.

-1.1831	confidence interval 99.% lower
-0.7844	confidence interval 99.% upper
0.1993	margin of error

The 99% confidence interval for the difference in level of bitterness is between -1.1831 and -0.7844.

1. [1 Mark] Describe in one sentence how the hypothesis test in part b) and the confidence interval in part c) reflect the relationship between confidence intervals and hypothesis tests as introduced in class.

Since the confidence interval does not contain 0, we can conclude that there is a real difference in the mean level of bitterness of these two types of beer. Thus, the hypothesis test in part b) and the confidence interval in part c) reflect the relationship between confidence intervals and hypothesis tests.

1. [7 Marks] The manager of the brewery thinks that assuming unequal population variances is wrong. Repeat part b), now manually, assuming equal population variances. Show your manual calculations and clearly state your conclusion. (Hint. You may use any statistics you need from part b)).

The hypothesis being tested is:

H₀: µ₁ = µ₂

H₁: µ₁ ≠ µ₂

Midnight Ottawa	Midnight Montreal
10.548	11.532	mean
0.261	0.216	std. dev.
23	19	n
	40	df
	-0.9838	difference (Midnight Ottawa - Midnight Montreal)
	0.0585	pooled variance
	0.2418	pooled std. dev.
	0.0750	standard error of difference
	0	hypothesized difference
	-13.124	t
	4.42E-16	p-value (two-tailed)

Since the p-value (0.0000) is less than the significance level (0.01), we can reject the null hypothesis.

Therefore, we can conclude that there is a real difference in the mean level of bitterness of these two types of beer.