Question

5. (a)A sample of 12 of bags of Calbie Chips were weighed (to the nearest gram), and listed, here as follows.

219, 226, 217, 224, 223, 216, 221, 228, 215, 229, 225, 229 Find a 95% confidence interval for the mean mass of bags of Calbie Chips.

[9 marks]

(b) Professor GeniusAtCalculus has two lecture sections (A and B) of the same 4th year Advanced Calculus (AMA 4301) course in Semester 2. She wants to investigate whether section A students maybe ”smarter” than section B students by comparing their perfor- mances in the midterm test. A random sample of 12 students were taken from section A, with mean midterm test score of 78.8 and standard deviation 8.5; and a random sample of 9 students were taken from section B, with mean midterm test score of 86 and standard deviation 9.3. Assume the population standard deviations of midterm test scores for both sections are the same. Construct the 90% confidence interval for the difference in midterm test scores of the two sections. Based on the sample midterm test scores from the two sections, can Professor GeniusAtCalculus conclude that there is any evidence that one section of students are ”smarter” than the other section? Justify your conclusions.

[8 marks]

(c) The COVID-19 (coronavirus) mortality rate of a country is defined as the ratio of the number of deaths due to COVID-19 divided by the number of (confirmed) cases of COVID-19 in that country. Suppose we want to investigate if there is any difference between the COVID-19 mortality rate in the US and the UK. On April 18, 2020, out of a sample of 671,493 cases of COVID-19 in the US, there was 33,288 deaths; and out of a sample of 109,754 cases of COVID-19 in the UK, there was 14,606 deaths. What is the 92% confidence interval in the true difference in the mortality rates between the two countries? What can you conclude about the difference in the mortality rates between the US and the UK? Justify your conclusions. [8 marks]

Answer 1

a ) using excel>addin>phstat>confidence interval

we have

Confidence Interval Estimate for the Mean
Data
Sample Standard Deviation	5.033222957
Sample Mean	222.6666667
Sample Size	12
Confidence Level	95%
Intermediate Calculations
Standard Error of the Mean	1.452966315
Degrees of Freedom	11
t Value	2.2010
Interval Half Width	3.1980
Confidence Interval
Interval Lower Limit	219.47
Interval Upper Limit	225.86

95% confidence interval for the mean mass of bags of Calbie Chips is (219.47,225.86)

2 ) using excel>addin>phstat>two sample test

we have

Pooled-Variance t Test for the Difference Between Two Means
(assumes equal population variances)
Data		Confidence Interval Estimate
Hypothesized Difference	0		for the Difference Between Two Means
Level of Significance	0.05
Population 1 Sample			Data
Sample Size	12		Confidence Level	90%
Sample Mean	78.8
Sample Standard Deviation	8.5		Intermediate Calculations
Population 2 Sample			Degrees of Freedom	19
Sample Size	9		t Value	1.7291
Sample Mean	86		Interval Half Width	46.2762
Sample Standard Deviation	93
			Confidence Interval
Intermediate Calculations		Interval Lower Limit	-53.4762
Population 1 Sample Degrees of Freedom	11		Interval Upper Limit	39.0762
Population 2 Sample Degrees of Freedom	8
Total Degrees of Freedom	19
Pooled Variance	3683.5132
Standard Error	26.7626
Difference in Sample Means	-7.2000
t Test Statistic	-0.2690
Two-Tail Test
Lower Critical Value	-2.0930
Upper Critical Value	2.0930
p-Value	0.7908

the 90% confidence interval for the difference in midterm test scores of the two sections is (-53.4762,39.0762)

since confidence interval contains 0 so we do not have sufficient evidence to conclude that that one section of students are ”smarter” than the other section.

Ans 3 ) using excel>addin>phstat>two sample test >z test for difference of proportion

we have

Z Test for Differences in Two Proportions
Data			Confidence Interval Estimate
Hypothesized Difference	0		of the Difference Between Two Proportions
Level of Significance	0.05
US			Data
Number of Items of Interest	33288		Confidence Level	92%
Sample Size	671493
UK			Intermediate Calculations
Number of Items of Interest	14606		Z Value	-1.7507
Sample Size	109754		Std. Error of the Diff. between two Proportions	0.0011
			Interval Half Width	0.0019
Intermediate Calculations
Group 1 Proportion	0.049573115		Confidence Interval
Group 2 Proportion	0.133079432		Interval Lower Limit	-0.0854
Difference in Two Proportions	-0.08350632		Interval Upper Limit	-0.0817
Average Proportion	0.0613
Z Test Statistic	-106.9171
Two-Tail Test
Lower Critical Value	-1.9600
Upper Critical Value	1.9600
p-Value	0.0000
Reject the null hypothesis

we will conclude that the difference in the mortality rates in  US is lower than UK becasue confidence interval contain only negative numbers the 92% confidence interval in the true difference in the mortality rates between the two countries is (-0.0854 ,-0.0817)