Question

For each of the following three sample sizes, con- struct the 95% confidence interval. Use a sample proportion of 0.40 throughout. What happens to interval width as sample size increases? Why? P. = 0.40 Sample A: N = 100 Sample B: N = 1000 Sample C: N = 10,000

Answer 1

Answer:

C= 95%, $\hat p$ = 0.40,

1)

for n= 100

formula for confidence interval is

p*(1-P) pZc* v

Where Zc is the z critical value for Zc = 1.96

0.40 + (1-0.40) 0.40 +1.96 */ 100

0.304 < P < 0.496

thus we get confidence interval = ( 0.304 ,  0.496)

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2)

for n = 1000

formula for confidence interval is

p*(1-P) pZc* v

Where Zc is the z critical value for Zc = 1.96

0.40 + (1-0.40) 0.40 +1.96 */ 1000

0.370 < P < 0.430

thus we get confidence interval = ( 0.370 , 0.430)

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3)

for n = 10000

formula for confidence interval is

p*(1-P) pZc* v

Where Zc is the z critical value for Zc = 1.96

0.40 + (1-0.40) 0.40 +1.96 */ 10000

0.390 < P < 0.410

thus we get confidence interval = ( 0.390 , 0.410)

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Width of confidence interval decreases as the sample size increases.

because, as sample size increases the standard error is decreases.

Hence, Width of confidence interval decreases.