Question

User Before the election, a poll of 60 voters found the proportion who support the Green candidate to be 25%. Calculate the 90% confidence interval for the population parameter. (Give your answers as a PERCENTAGE rounded to TWO DECIMAL PLACES: exclude any trailing zeros and DO NOT INSERT THE % SIGN) Give the lower limit of the 90% confidence interval Give the upper limit of the 90% confidence interval

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Maude

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34 Answers

To calculate the 90% confidence interval for the population parameter, we can use the formula:

\text{Confidence Interval} = \text{Sample Proportion} \pm \text{Critical Value} \times \text{Standard Error}

where:

- Sample Proportion is the proportion of voter support found in the poll (p̂)

- Critical Value is the value from the standard normal distribution corresponding to the desired confidence level (Z)

- Standard Error is the standard deviation of the sample proportion (σp̂)

Step 1: Calculate the Sample Proportion (p̂)

Given that the proportion of voters supporting the Green candidate is 25%, we can say that p̂ = 0.25.

Step 2: Calculate the Critical Value (Z)

Since we want a 90% confidence interval, we need to find the critical value that corresponds to a 5% margin of error on each side of the mean. Using a standard normal distribution table or calculator, we find that the critical value Z for a 90% confidence level is approximately 1.645.

Step 3: Calculate the Standard Error (σp̂)

The standard error of the sample proportion can be calculated using the formula:

\sigma_{p̂} = \sqrt{\frac{p̂(1-p̂)}{n}}

where n is the sample size. In this case, the sample size is 60.

\sigma_{p̂} = \sqrt{\frac{0.25(1-0.25)}{60}}

\sigma_{p̂} = \sqrt{\frac{0.1875}{60}}

\sigma_{p̂}\approx0.0559

Step 4: Calculate the Confidence Interval

Now we can plug in the values into the confidence interval formula:

\text{Confidence Interval}=0.25\pm1.645\times0.0559

Step 5: Calculate the Lower Limit

\text{Lower Limit}=0.25-1.645\times0.0559=0.25-0.0919555=0.1580445\approx15.80\%

Step 6: Calculate the Upper Limit

\text{Upper Limit}=0.25+1.645\times0.0559=0.25+0.0919555=0.3419555\approx34.20\%

Answer:

The lower limit of the 90% confidence interval is approximately 15.80%

The upper limit of the 90% confidence interval is approximately 34.20%

where:

- Sample Proportion is the proportion of voter support found in the poll (p̂)

- Critical Value is the value from the standard normal distribution corresponding to the desired confidence level (Z)

- Standard Error is the standard deviation of the sample proportion (σp̂)

Step 1: Calculate the Sample Proportion (p̂)

Given that the proportion of voters supporting the Green candidate is 25%, we can say that p̂ = 0.25.

Step 2: Calculate the Critical Value (Z)

Since we want a 90% confidence interval, we need to find the critical value that corresponds to a 5% margin of error on each side of the mean. Using a standard normal distribution table or calculator, we find that the critical value Z for a 90% confidence level is approximately 1.645.

Step 3: Calculate the Standard Error (σp̂)

The standard error of the sample proportion can be calculated using the formula:

where n is the sample size. In this case, the sample size is 60.

Step 4: Calculate the Confidence Interval

Now we can plug in the values into the confidence interval formula:

Step 5: Calculate the Lower Limit

Step 6: Calculate the Upper Limit

Answer:

The lower limit of the 90% confidence interval is approximately 15.80%

The upper limit of the 90% confidence interval is approximately 34.20%

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