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To estimate the percentage of a state's voters who support the current governor for reelection, three newspapers each survey a simple random sample of voters. Each paper calculates the percentage of voters in their sample who support the governor and uses that as an estimate for the population parameter. Here are the results:

- The Tribune: [tex]n=600[/tex] voters sampled; sample estimate [tex]=58 \%[/tex]
- The Herald: [tex]n=400[/tex] voters sampled; sample estimate [tex]=54 \%[/tex]
- The Times: [tex]n=200[/tex] voters sampled; sample estimate [tex]=68 \%[/tex]

All else being equal, which newspaper's estimate is likely closest to the actual percentage of voters who support the governor for reelection?

A. The Times, at [tex]68 \%[/tex]
B. The Herald, at [tex]54 \%[/tex]
C. The Tribune, at [tex]58 \%[/tex]



Answer :

GinnyAnswer
To determine which newspaper's estimate is likely closest to the actual percentage of voters who support the governor for reelection, we need to compute the standard error for each newspaper's estimate. The standard error provides a measure of how much the sample estimate is expected to fluctuate due to the randomness inherent in sampling. A smaller standard error indicates a more reliable estimate.

The formula for the standard error (SE) for a proportion is given by:

[tex]\[ SE = \sqrt{\frac{p(100 - p)}{n}}, \][/tex]

where [tex]\( p \)[/tex] is the sample estimate percentage and [tex]\( n \)[/tex] is the sample size.

Let's compute the standard error for each newspaper:

1. The Tribune:
- Sample estimate ([tex]\( p \)[/tex]) = 58%
- Sample size ([tex]\( n \)[/tex]) = 600

[tex]\[ SE_{\text{Tribune}} = \sqrt{\frac{58 \times (100 - 58)}{600}} \approx 2.0149441679609885 \][/tex]

2. The Herald:
- Sample estimate ([tex]\( p \)[/tex]) = 54%
- Sample size ([tex]\( n \)[/tex]) = 400

[tex]\[ SE_{\text{Herald}} = \sqrt{\frac{54 \times (100 - 54)}{400}} \approx 2.4919871588754225 \][/tex]

3. The Times:
- Sample estimate ([tex]\( p \)[/tex]) = 68%
- Sample size ([tex]\( n \)[/tex]) = 200

[tex]\[ SE_{\text{Times}} = \sqrt{\frac{68 \times (100 - 68)}{200}} \approx 3.2984845004941286 \][/tex]

Now we compare the standard errors:

- SE_{\text{Tribune}} \approx 2.0149441679609885
- SE_{\text{Herald}} \approx 2.4919871588754225
- SE_{\text{Times}} \approx 3.2984845004941286

The Tribune's estimate has the smallest standard error, which implies that its estimate of 58% is likely the most reliable and closest to the actual percentage of voters who support the governor for reelection.

Therefore, the newspaper's estimate that is likely closest to the actual percentage is:

C. The Tribune, at 58%

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