Question

An education minister would like to know whether students at Gedrassi high school on average perform better at English or at Mathematics. Denoting by μ1 the mean score for all Gedrassi students in a standardized English exam and μ2 the mean score for all Gedrassi students in a standardized Mathematics exam, the minister would like to get a 95% confidence interval estimate for the difference between the means: μ1 - μ2.

A study was conducted where many students were given a standardized English exam and a standardized Mathematics exam and their pairs of scores were recorded. Unfortunately, most of the data has been misplaced and the minister only has access to scores for 4 students.

Student	English	Mathematics
Student 1	80	66
Student 2	75	70
Student 3	75	66
Student 4	76	66

The populations of test scores are assumed to be normally distributed. The minister decides to construct the confidence interval with these 4 pairs of data points. This Student's t distribution table may assist you in answering the following questions.

a)Calculate the lower bound for the confidence interval. Give your answer to 3 decimal places.

Lower bound =

b)Calculate the upper bound for the confidence interval. Give your answer to 3 decimal places.

Upper bound =

An assistant claims that there is no difference between the average English score and the average Math score for a student at Gedrassi high school.

c)Based on the confidence interval the minister constructs, the claim by the assistant can or cannot be ruled out.

Answer 1

table for necessary calculation:-

english(x)	maths(y)	(x- x bar)^2	(y-y bar)^2
80	66	12.25	1
75	70	2.25	9
75	66	2.25	1
76	66	0.25	1
sum=306	sum=268	sum=17	sum=12

so , we have,

according to the above formula df = 5

t value for df=5 , alpha= 0.05 be:-

2.571

so,

a) lower bound for the confidence interval be:-

b).upper bound for the confidence interval be:-

c). hypothesis:-

our 95 % ci is= (5.503 , 13.497)

since the confidence interval do not contain the hypothesized difference , that is 0 , we reject the null hypothesis.

so, we conclude that

Based on the confidence interval the minister constructs, the claim by the assistant  can not be supported.

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