Question

A national television network took an exit poll of 1350 voters after each had cast a vote in a state gubernatorial election. Of? them, 610 said they voted for the Republican candidate and 740 said they voted for the Democratic candidate. Treating the sample as a random sample from the population of all? voters, a 95?% confidence interval for the proportion of all voters voting for the Democratic candidate was (0.522,0.575). Suppose the same proportions resulted from nequals=135 ?(instead of1350?), with counts of 61 and 74?, and that there are only two candidates. Complete parts a and b below.

**a.** Does a 95?% confidence interval using the
smaller sample size allow you to predict the? winner? Explain.

The 95?% confidence interval for the proportion of all voters voting for the Democratic candidate is ( , ). Now a 95?% confidence interval (does or does not?) allow you to predict the? winner,

since this interval (includes, does not include?) (1, 0.5, 0?)

(Round to three decimal places as? needed.)

**b.** Explain why the same proportions but with
smaller samples provide less information.? (Hint: What effect does
n have on the standard? error?)

A smaller sample results in a (greater, lesser?) standard? error, which results in a (greater, lesser?) (point estimate, margin of error?) for the same proportions and confidence? level, meaning less information is provided.

Answer #1

**a.** p = 74/135 = 0.548

95% confidence interval using the smaller sample size is given as :

Lower limit = p - z_{0.025} (p*(1-p) / n)^{0.5}
= 0.548 - 1.96 *(0.548*(1 - 0.548) / 135)^{0.5} =
**0.464**

Upper limit = p + z_{0.025} (p*(1-p) /
n)^{0.5} = 0.548 + 1.96 *(0.548*(1 - 0.548) /
135)^{0.5} = **0.632**

The 95% confidence interval for the proportion of all voters
voting for the Democratic candidate is **(**
**0.464 , 0.632** **)**

Now a 95% confidence interval **does not** allow
you to predict the winner, since this interval **includes**
**0.5**

**b.**

A smaller sample results in a **greater**
standard error, which results in a **greater margin of error** for
the same proportions and confidence level, meaning less information
is provided.

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