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Answer :
Answer:
A.) 54 subjects
B.) 216 subjects
C.) doubling the accuracy results in 4 times the sample size
D.) 38 subjects
Decreasing confidence level decreases sample size. For fixed error margin, the lower the confidence level, the lower the sample size
Explanation:
Standard deviation (s) = 7.5 hours
A.)
Error margin 'E' = 2
Confidence level = 0.95
α = 1 - 0.95 = 0.05, α/2 = 0.025
Z - value at α/2 = 0.025 = 1.96
Sample size = [(1.96 × 7.5)/2]^2
Sample size = 7.35^2 = 54.022
54 subjects
B.) E = 1
Sample size = [(1.96 × 7.5)/1]^2
Sample size = 14.7^2 = 216.09
216 subjects
C.) from the above, doubling the accuracy results in 4 times the sample size.
D.) Using a confidence interval of 90%
Error margin 'E' = 2
Confidence level = 0.90
α = 1 - 0.90 = 0.1, α/2 = 0.05
Z - value at α/2 = 0.05 = 1.645
Sample size = [(1.645 × 7.5)/2]^2
Sample size = 6.16875^2 = 38.05
=38 subjects
Decreasing confidence level decreases sample size. For fixed error margin, the lower the confidence level, the lower the sample size
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Final answer:
To estimate the mean number of hours people watch television, a formula involving the Z-score, standard deviation, and margin of error is used to calculate the necessary sample size. With a 95% confidence level, 35 people are needed for a 2-hour margin and 137 people for a 1-hour margin. Reducing accuracy or confidence level decreases sample size requirements.
Explanation:
To estimate the mean number of hours a person watches television using confidence intervals, we can use the formula for sample size:
n = (Z·s/E)²
where n is the sample size, Z is the Z-score corresponding to the confidence level, s is the sample standard deviation, and E is the margin of error.
(a) 95% Confidence, 2 hours margin of error
Z = Z-score for 95% confidence approximately equals 1.96. Using the given standard deviation (s = 7.5),
n = (1.96· 7.5/2)² ≈ 34.22
Therefore, approximately 35 people are needed. We round up since the sample size must be a whole number.
(b) 95% Confidence, 1 hour margin of error
Z stays the same, but the margin of error (E) is now 1 hour:
n = (1.96· 7.5/1)² ≈ 136.89
Thus, about 137 people are needed.
(c) Effect of Doubling Accuracy on Sample Size
The required sample size increases by a factor of 4 when the margin of error is halved, demonstrating that precision is costly in terms of sample size.
(d) 90% Confidence, 2 hours margin of error
Z = Z-score for 90% confidence approximately equals 1.645. With the margin of error back at 2 hours:
n = (1.645· 7.5/2)² ≈ 26.49
Approximately 27 people are needed. The sample size decreases with a lower confidence level because there is less certainty that the interval contains the true mean.
