Guided Practice: Confidence Intervals, Proportions, and Sample Size in Statistics

Q1. A survey of 100 fatal accidents showed that in 44 cases the driver at fault was inadequately insured. Find a point estimate for p, the population proportion of accidents where the driver at fault was inadequately insured.

Background

Topic: Point Estimation of Population Proportion

This question tests your ability to calculate a point estimate for a population proportion based on sample data.

Key Terms and Formula:

• Point Estimate: A single value estimate for a population parameter (here, the proportion p).

• Sample Proportion (\( \hat{p} \)): \( \hat{p} = \frac{x}{n} \)

• Where:

  • \( x \) = number of successes (cases with inadequately insured drivers)

  • \( n \) = total number of cases in the sample

Step-by-Step Guidance

1. Identify the sample size (n) and the number of successes (x) from the problem statement.

2. Write the formula for the sample proportion: \( \hat{p} = \frac{x}{n} \)

3. Substitute the values for x and n into the formula.

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Q2. Determine the critical value z\( \alpha/2 \) that corresponds to the given level of confidence: 92%.

Background

Topic: Critical Values for Confidence Intervals

This question tests your understanding of how to find the z-score that corresponds to a specified confidence level for constructing confidence intervals.

Key Terms and Formula:

• Critical Value (z\( \alpha/2 \)): The z-score that marks the boundaries of the middle (confidence level) percent of the standard normal distribution.

• Confidence Level (CL): The probability that the interval estimate contains the population parameter.

• \( \alpha = 1 - \text{Confidence Level} \)

• \( z_{\alpha/2} \) is found such that the area between \( -z_{\alpha/2} \) and \( z_{\alpha/2} \) is equal to the confidence level.

Step-by-Step Guidance

1. Convert the confidence level to a decimal (e.g., 92% = 0.92).

2. Calculate \( \alpha = 1 - \text{confidence level} \).

3. Divide \( \alpha \) by 2 to get \( \alpha/2 \).

4. Use the standard normal (z) table to find the z-score with a cumulative area of \( 1 - \alpha/2 \).

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Q3. Assume that a sample is used to estimate a population proportion p. Find the margin of error E that corresponds to the given statistics and confidence level. Round the margin of error to four decimal places. 95% confidence; the sample size is 5700, of which 20% are successes.

Background

Topic: Margin of Error for Proportion Estimates

This question tests your ability to calculate the margin of error for a sample proportion at a given confidence level.

Key Terms and Formula:

• Margin of Error (E): The maximum expected difference between the true population parameter and a sample estimate.

• \( E = z_{\alpha/2} \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} } \)

• Where:

  • \( \hat{p} \) = sample proportion

  • n = sample size

  • \( z_{\alpha/2} \) = critical value for the confidence level

Step-by-Step Guidance

1. Identify the sample proportion (\( \hat{p} \)), sample size (n), and confidence level.

2. Find the critical value \( z_{\alpha/2} \) for a 95% confidence level.

3. Plug the values into the margin of error formula: \( E = z_{\alpha/2} \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} } \)

4. Calculate the value under the square root, but do not compute the final value yet.

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Q4. When 410 junior college students were surveyed, 185 said they have a passport. Construct a 95% confidence interval for the proportion of junior college students that have a passport. Round to the nearest thousandth.

Background

Topic: Confidence Interval for a Population Proportion

This question tests your ability to construct a confidence interval for a population proportion using sample data.

Key Terms and Formula:

• Sample Proportion (\( \hat{p} \)): \( \hat{p} = \frac{x}{n} \)

• Confidence Interval: \( \hat{p} \pm z_{\alpha/2} \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} } \)

Step-by-Step Guidance

1. Calculate the sample proportion \( \hat{p} \) using the given data.

2. Find the critical value \( z_{\alpha/2} \) for a 95% confidence level.

3. Compute the standard error: \( \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} } \)

4. Set up the confidence interval formula using the values from the previous steps.

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Q5. A researcher at a major clinic wishes to estimate the proportion of the adult population of the United States that has sleep deprivation. How large a sample is needed in order to be 98% confident that the sample proportion will not differ from the true proportion by more than 5%?

Background

Topic: Determining Sample Size for Proportion Estimates

This question tests your ability to determine the minimum sample size required to estimate a population proportion with a specified margin of error and confidence level.

Key Terms and Formula:

• Sample Size (n): \( n = \frac{ z_{\alpha/2}^2 \cdot p \cdot q }{ E^2 } \)

• Where:

  • \( p \) = estimated population proportion (if unknown, use 0.5 for maximum variability)

  • \( q = 1 - p \)

  • \( E \) = desired margin of error

  • \( z_{\alpha/2} \) = critical value for the confidence level

Step-by-Step Guidance

1. Identify the desired confidence level (98%) and margin of error (5% or 0.05).

2. Since p is unknown, use p = 0.5 and q = 0.5 for the most conservative estimate.

3. Find the critical value \( z_{\alpha/2} \) for a 98% confidence level.

4. Plug the values into the sample size formula and simplify, but do not compute the final value yet.

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Q6. Express the confidence interval 0.148 < p < 0.652 in the form of p ± E.

Background

Topic: Confidence Interval Notation

This question tests your ability to convert a confidence interval from the inequality form to the p ± E form.

Key Terms and Formula:

• Confidence Interval (Inequality Form): \( a < p < b \)

• Confidence Interval (p ± E Form): \( p \pm E \), where \( p = \frac{a + b}{2} \) and \( E = \frac{b - a}{2} \)

Step-by-Step Guidance

1. Identify the lower (a) and upper (b) bounds of the interval.

2. Calculate the midpoint: \( p = \frac{a + b}{2} \)

3. Calculate the margin of error: \( E = \frac{b - a}{2} \)

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Q7. Express the confidence interval 0.469 ± 0.077 in the form of p - E < p < p + E.

Background

Topic: Confidence Interval Notation

This question tests your ability to convert a confidence interval from the p ± E form to the inequality form.

Key Terms and Formula:

• Confidence Interval (p ± E Form): \( p \pm E \)

• Confidence Interval (Inequality Form): \( p - E < p < p + E \)

Step-by-Step Guidance

1. Identify the point estimate (p) and margin of error (E) from the given expression.

2. Calculate the lower bound: \( p - E \)

3. Calculate the upper bound: \( p + E \)

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Q8. Use the given data to find the minimum sample size required to estimate the population proportion. Margin of error: 0.008; confidence level: 99%; p and q unknown.

Background

Topic: Sample Size for Proportion Estimates (Unknown p and q)

This question tests your ability to determine the minimum sample size needed for a specified margin of error and confidence level when the population proportion is unknown.

Key Terms and Formula:

• Sample Size (n): \( n = \frac{ z_{\alpha/2}^2 \cdot p \cdot q }{ E^2 } \)

• When p and q are unknown, use p = 0.5 and q = 0.5.

Step-by-Step Guidance

1. Identify the margin of error (E), confidence level, and note that p and q are unknown.

2. Use p = 0.5 and q = 0.5 for maximum variability.

3. Find the critical value \( z_{\alpha/2} \) for a 99% confidence level.

4. Plug the values into the sample size formula and simplify, but do not compute the final value yet.

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Q9. Margin of error: 0.02; confidence level: 95%; from a prior study, p is estimated by the decimal equivalent of 52%. Find the minimum sample size required to estimate the population proportion.

Background

Topic: Sample Size for Proportion Estimates (Known p)

This question tests your ability to determine the minimum sample size needed for a specified margin of error and confidence level when an estimate for p is available.

Key Terms and Formula:

• Sample Size (n): \( n = \frac{ z_{\alpha/2}^2 \cdot p \cdot q }{ E^2 } \)

• p = 0.52, q = 1 - p = 0.48

Step-by-Step Guidance

1. Identify the margin of error (E), confidence level, and the estimated value of p.

2. Calculate q as 1 - p.

3. Find the critical value \( z_{\alpha/2} \) for a 95% confidence level.

4. Plug the values into the sample size formula and simplify, but do not compute the final value yet.

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Q10. How many women must be randomly selected to estimate the mean weight of women in one age group? We want 90% confidence that the sample mean is within 2.7 lb of the population mean, and the population standard deviation is known to be 22 lb.

Background

Topic: Sample Size for Estimating a Population Mean (Known Standard Deviation)

This question tests your ability to determine the minimum sample size required to estimate a population mean with a specified margin of error and confidence level, given a known population standard deviation.

Key Terms and Formula:

• Sample Size (n): \( n = \left( \frac{ z_{\alpha/2} \cdot \sigma }{ E } \right)^2 \)

• Where:

  • \( \sigma \) = population standard deviation

  • \( E \) = desired margin of error

  • \( z_{\alpha/2} \) = critical value for the confidence level

Step-by-Step Guidance

1. Identify the confidence level, margin of error, and population standard deviation.

2. Find the critical value \( z_{\alpha/2} \) for a 90% confidence level.

3. Plug the values into the sample size formula and simplify, but do not compute the final value yet.

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Q11. Use the confidence level and sample data to find a confidence interval for estimating the population mean µ. Test scores: n = 104, x̄ = 95.3, σ = 6.5; 99% confidence.

Background

Topic: Confidence Interval for a Population Mean (Known Standard Deviation)

This question tests your ability to construct a confidence interval for a population mean when the population standard deviation is known.

Key Terms and Formula:

• Confidence Interval: \( \bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}} \)

• Where:

  • \( \bar{x} \) = sample mean

  • \( \sigma \) = population standard deviation

  • n = sample size

  • \( z_{\alpha/2} \) = critical value for the confidence level

Step-by-Step Guidance

1. Identify the sample mean, standard deviation, sample size, and confidence level.

2. Find the critical value \( z_{\alpha/2} \) for a 99% confidence level.

3. Calculate the standard error: \( \frac{\sigma}{\sqrt{n}} \)

4. Set up the confidence interval formula using the values from the previous steps.

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Q12. A random sample of 187 full-grown lobsters had a mean weight of 19 ounces. Assume the population standard deviation is 3.3 ounces. Construct a 98% confidence interval for the population mean µ.

Background

Topic: Confidence Interval for a Population Mean (Known Standard Deviation)

This question tests your ability to construct a confidence interval for a population mean when the population standard deviation is known.

Key Terms and Formula:

• Confidence Interval: \( \bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}} \)

Step-by-Step Guidance

1. Identify the sample mean, standard deviation, sample size, and confidence level.

2. Find the critical value \( z_{\alpha/2} \) for a 98% confidence level.

3. Calculate the standard error: \( \frac{\sigma}{\sqrt{n}} \)

4. Set up the confidence interval formula using the values from the previous steps.

Try solving on your own before revealing the answer!

Q13. Use the given degree of confidence and sample data to construct a confidence interval for the population mean µ. Assume that the population has a normal distribution. n = 12, x̄ = 23.6, s = 6.6, 99% confidence.

Background

Topic: Confidence Interval for a Population Mean (Unknown Standard Deviation, Small Sample)

This question tests your ability to construct a confidence interval for a population mean when the population standard deviation is unknown and the sample size is small (n < 30), requiring the use of the t-distribution.

Key Terms and Formula:

• Confidence Interval: \( \bar{x} \pm t_{\alpha/2, df} \frac{s}{\sqrt{n}} \)

• Where:

  • \( \bar{x} \) = sample mean

  • \( s \) = sample standard deviation

  • n = sample size

  • df = degrees of freedom = n - 1

  • \( t_{\alpha/2, df} \) = critical value from the t-distribution

Step-by-Step Guidance

1. Identify the sample mean, sample standard deviation, sample size, and confidence level.

2. Calculate the degrees of freedom (df = n - 1).

3. Find the critical value \( t_{\alpha/2, df} \) for a 99% confidence level and the calculated degrees of freedom.

4. Calculate the standard error: \( \frac{s}{\sqrt{n}} \)

5. Set up the confidence interval formula using the values from the previous steps.

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Q14. A laboratory tested twelve chicken eggs and found that the mean amount of cholesterol was 185 milligrams with s = 17.6 milligrams. Construct a 95% confidence interval for the true mean cholesterol content of all such eggs.

Background

Topic: Confidence Interval for a Population Mean (Unknown Standard Deviation, Small Sample)

This question tests your ability to construct a confidence interval for a population mean when the population standard deviation is unknown and the sample size is small, requiring the use of the t-distribution.

Key Terms and Formula:

• Confidence Interval: \( \bar{x} \pm t_{\alpha/2, df} \frac{s}{\sqrt{n}} \)

Step-by-Step Guidance

1. Identify the sample mean, sample standard deviation, sample size, and confidence level.

2. Calculate the degrees of freedom (df = n - 1).

3. Find the critical value \( t_{\alpha/2, df} \) for a 95% confidence level and the calculated degrees of freedom.

4. Calculate the standard error: \( \frac{s}{\sqrt{n}} \)

5. Set up the confidence interval formula using the values from the previous steps.

Try solving on your own before revealing the answer!