Questions: A researcher wishes to estimate, with 95% confidence, the population proportion of likely U.S. voters who think Congress is doing a good or excellent job. Her estimate must be accurate within 3% of the true proportion. (a) No preliminary estimate is available. Find the minimum sample size needed. (b) Find the minimum sample size needed, using a prior study that found that 40% of the respondents said they think Congress is doing a good or excellent job. (c) Compare the results from parts (a) and (b). (a) What is the minimum sample size needed assuming that no prior information is available? n= (Round up to the nearest whole number as needed.)

A researcher wishes to estimate, with 95% confidence, the population proportion of likely U.S. voters who think Congress is doing a good or excellent job. Her estimate must be accurate within 3% of the true proportion.
(a) No preliminary estimate is available. Find the minimum sample size needed.
(b) Find the minimum sample size needed, using a prior study that found that 40% of the respondents said they think Congress is doing a good or excellent job.
(c) Compare the results from parts (a) and (b).
(a) What is the minimum sample size needed assuming that no prior information is available?
n= (Round up to the nearest whole number as needed.)

Transcript text: A researcher wishes to estimate, with $95 \%$ confidence, the population proportion of likely U.S. voters who think Congress is doing a good or excellent job. Her estimate must be accurate within $3 \%$ of the true proportion. (a) No preliminary estimate is available. Find the minimum sample size needed. (b) Find the minimum sample size needed, using a prior study that found that $40 \%$ of the respondents said they think Congress is doing a good or excellent job. (c) Compare the results from parts (a) and (b). (a) What is the minimum sample size needed assuming that no prior information is available? $n=$ $\square$ (Round up to the nearest whole number as needed.)

Solution

Solution Steps

Step 1: Identify the Parameters

Preliminary estimate of population proportion ($\hat{p}$): 0.5
Margin of Error ($E$): 0.03
Confidence Level ($CL$): 95%
Z-score ($Z$): 1.96

Step 2: Apply the Formula for Minimum Sample Size

The formula to calculate the minimum sample size is: \[n = \left(\frac{Z^2 \cdot \hat{p} \cdot (1 - \hat{p})}{E^2}\right)\] Substituting the values we have: \[n = \left(\frac{1.96^2 \cdot 0.5 \cdot (1 - 0.5)}{0.03^2}\right)\] \[n = 1067.111\]

Step 3: Rounding Up

The sample size must be a whole number. Therefore, we round up the calculated sample size. Rounded sample size ($n$): 1068

Final Answer:

The minimum sample size needed is ^1068^ (rounded to the nearest whole number). This ensures that the estimate of the population proportion is within a margin of error of 0.03 with a 95% confidence level.

Step 1: Identify the Parameters

Preliminary estimate of population proportion ($\hat{p}$): 0.4
Margin of Error ($E$): 0.03
Confidence Level ($CL$): 95%
Z-score ($Z$): 1.96

Step 2: Apply the Formula for Minimum Sample Size

The formula to calculate the minimum sample size is: \[n = \left(\frac{Z^2 \cdot \hat{p} \cdot (1 - \hat{p})}{E^2}\right)\] Substituting the values we have: \[n = \left(\frac{1.96^2 \cdot 0.4 \cdot (1 - 0.4)}{0.03^2}\right)\] \[n = 1024.427\]

Step 3: Rounding Up

The sample size must be a whole number. Therefore, we round up the calculated sample size. Rounded sample size ($n$): 1025

Final Answer:

The minimum sample size needed is ^1025^ (rounded to the nearest whole number). This ensures that the estimate of the population proportion is within a margin of error of 0.03 with a 95% confidence level.

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