Question

A travel magazine wanted to estimate the mean amount of leisure time per week enjoyed by adults. The research department at the magazine took a sample of 36 adults and obtained the following data on the weekly leisure time (in hours):

15 12 18 23 11 21 16 13    9 19 26 14    7 18 11 15 23 26

10    8 17 21 12    7 19 21 11 13 21 16 14    9 15 12 10 14

a. What is the point estimate of the mean leisure time per week enjoyed by all adults?

b. Construct a 99% confidence interval for the mean leisure time per week enjoyed by all adults.

c. What sample size would you have to collect so that the margin of error (E) is 1 hour/week for a 99% confidence interval? Assume that the sample standard deviation is true for the population.

d. For alpha=.01, test the statement that the mean amount of leisure time per week enjoyed by adults in the sample of 36 adults is less than 18 hours/week.

Answer 1

Mean X̅ = Σ Xi / n
X̅ = 547 / 36 = 15.1944

Sample Standard deviation SX = √ ( (Xi - X̅ )2 / n - 1 )
SX = √ ( 953.6392 / 36 -1 ) = 5.2199

Part a)

Point Estimate   X̅ = 15.1944

Part b)

Confidence Interval
X̅ ± t(α/2, n-1) S/√(n)
t(α/2, n-1) = t(0.01 /2, 36- 1 ) = 2.724
15.1944 ± t(0.01/2, 36 -1) * 5.2199/√(36)
Lower Limit = 15.1944 - t(0.01/2, 36 -1) 5.2199/√(36)
Lower Limit = 12.8246
Upper Limit = 15.1944 + t(0.01/2, 36 -1) 5.2199/√(36)
Upper Limit = 17.5642
99% Confidence interval is ( 12.8246 , 17.5642 )

Part c)

Sample size can be calculated by below formula
n = (( Z(α/2) * σ) / e )2
n = (( Z(0.01/2) * 5.2199 ) / 1 )2
Critical value Z(α/2) = Z(0.01/2) = 2.5758
n = (( 2.5758 * 5.2199 ) / 1 )2
n = 181
Required sample size at 99% confident is 181.

Part d)

To Test :-

H0 :- µ = 18
H1 :- µ < 18

Test Statistic :-
t = ( X̅ - µ ) / (S / √(n) )
t = ( 15.1944 - 18 ) / ( 5.2199 / √(36) )
t = -3.2249

Test Criteria :-
Reject null hypothesis if t < -t(α, n-1)
Critical value t(α, n-1) = t(0.01 , 36-1) = 2.438
t < - t(α, n-1) = -3.2249 < - 2.438
Result :- Reject null hypothesis

There is sufficient evidnece to support the claim that µ < 18.