Question

Education officials in a particular state claim that their state is especially effective in preparing students for college. One
assessment of preparation for college is the mathematics part of the Scholastic Aptitude Test (SAT-Math). Suppose that one
wanted to infer the population mean SAT-Math score for students in this state using the mean of a simple random sample of
SAT-Math scores of students from this state, and one knew that the population standard deviation σ = 100‡.

Complete the table below with six confidence intervals. First, assume that the sample mean assigned to you came from a sample of 25 students, and find confidence intervals for the three assigned confidence levels. Then do this assuming that the sample mean came from a sample of 400 students. You can write the confidence intervals in any acceptable form. (Confidence level: e.g., 80%, 90%, etc.) n=25 n = 400 Confidence Level #1 → Confidence Level #2 → Confidence Level #3→

sample mean: 525

Confidence Level # 1: 90%

Confidence Level # 2: 96%

Confidence Level # 3: 99%

b) (i) What is the probability, assuming that μ = 500 and “knowing” that σ = 100, that one would obtain a mean from a
sample of 25 that is at least as high as the sample mean assigned to you? (This is the P-value for a test of the hypotheses
H0: μ = 500 and Ha: μ > 500.) (ii) With α = .05, say whether H0 should be rejected or retained, and why.

c) (i) What is the probability, assuming that μ = 500 and “knowing” that σ = 100, that one would obtain a mean from a
sample of 400 that is at least as high as the sample mean assigned to you? (This is the P-value for a test of the hypotheses
H0: μ = 500 and Ha: μ > 500.) (ii) With α = .05, say whether H0 should be rejected or retained, and why.

d) The confidence interval that you constructed in the middle cell of the right column of the above table (shaded in grey)
provides the information necessary to decide between H0: μ = 500 and Ha: μ ≠ 500 at some level of α. (i) What is the level
of α? (ii) With that level of α, given the sample mean assigned to you, should H0: μ = 500 be rejected? Why?

Answer 1

(a)

=> z-value to calulate the confidence interval :-

(1-alpha)*100% confidence interval for the population mean=sample mean±z(alpha/2)*sd/sqrt(n)

for n=25

	n=	25
	sample mean=	515
	sd=	100.00
		z-value	lower limit	upper limit
1	90% confidence interval	1.6	482.10	547.90
2	96% confidence interval	2.05	473.93	556.07
3	99% confidence interval	2.58	463.48	566.52

	n=	400
	sample mean=	515
	sd=	100.00
area		z-value	lower limit	upper limit
1	90% confidence interval	1.6	506.78	523.22
2	96% confidence interval	2.05	504.73	525.27
3	99% confidence interval	2.58	502.12	527.88

(b) standard normal variate z=(x-µ)/σ_x

for =515, σ=σ/sqrt(n)=100/sqrt(25)=20 and z=(-µ)/σ=(515-500)/20=0.75

P( sample mean assign)=P( >515)=P(Z>0.75)=1-P(Z<0.75)=1-0.7733=0.2267

(c)for =515, σ=σ/sqrt(n)=100/sqrt(400)=5 and z=(-µ)/σ=(515-500)/5=3

P(sample mean assign)=P( >515)=P(Z>3)=1-P(Z<0.75)=1-0.9987=0.0013