Question

You are working for an advocacy group as a summer intern and your assignment is to find the mean annual heating costs for at-risk families in the Boston area. You’d like a 95% confidence interval, and you want the margin of error to be $100. A preliminary study indicates that s will likely be about $500. a. Write a formula for the number of houses n needed to sample for the experiment. b. Estimate the formula—don’t worry about being too precise! c. It’s claimed that the average annual heating cost for at-risk families in the Boston area is $3,300. You believe it is less than that. Write H0 and Ha for a hypothesis test. d. You receive funding to survey 64 households about their heating costs, and you find that y = $3,000 and s = $80. Write a formula for the 99% CI. e. In the context of this problem, explain what a 99% CI is. f. Suppose you end up rejecting H0 and accepting Ha , and the true average annual heating costs for Boston-area at-risk families is $3,299. Have you committed an error? If so, what type?

Answer 1

Confidence Interval = 95%

Z -score, z = 1.96

Margin Of Error, ME = 100

Standard deviation, s = 500

a.)

The formula to find number of houses is given by the formula to find sample size,

n = (Z*s / Margin of Error)²

Here Z = 1.96, So,

n = 3.84 * (s / Margin of Error)²

b.)

n = 3.84 * ( 500/100)²

n = 3.84 * 25

n = 96

c.)

The Hypotheses are:

Ha: Average Heating Cost for House holds at risk is 3000$

Ha:   = 3300

Ha: Average Heating Cost for House holds at risk is less than 3000$

Ha:   < 3300

d.)

Given that n = 64

s = 84

y = 3000

For 99% CI,

Z = 2.58

Standard Error,

SE = s /

So, Confidence interval is given as:

CI = y Z * SE

CI = y 2.58 * SE

where SE = s /

e.)

The 99% CI implies that we can say with 99% confidence that heating cost for at-risk families will lie in range of Calculated Confidence Interval