Question

A researcher has conducted a survey using a simple random sample of 500 gold club members to create a confidence interval to estimate the proportion of golf club members favoring a 1% annual increase in dues to remodel the clubhouse assume that the sample proportion does not change the researcher now decides to survey a random sample of 125 people instead of 500 golf club members which of the following statments best describes how the confidence interval is affected by this change

a. The width of new interval is aobut t same width as the original interval
b. the width of the new interval is about twice the width of the original interval
c. the width of the new interval is about one hald the width of the original interval
d. the width of the new interval is about on fourth the width of the original interval
e. the width of the new interval is about 4 times the width of the original interval

Answer 1

Answer: B.

Procedure we follow in calculating confidence level for proportion.

i. standard error = sqrt ( (p*q) /n) )

ii. margin of error = z a/2 * (stanadard error)

where,

za/2 = z-table value

level of significance, alpha

from standard normal table, two tailed z alpha/2

iii. ci = [ p ± margin of error ]

with above, we see that sample size n is inversely proportional to the confidence level

i.e increasing the sample size narrow the confidence interval

& decreasing the sample size widen the confidence interval.

CI ~ sqrt(1/n)

size is cut down to 4 times, i.e 500/125 = 4 times decreased

CI ~ sqrt(1/(n/4))

CI ~ sqrt(4/(n)

CI ~ 2 * sqrt(1/(n)

from above, we conclude that it is twice the previous