Question

In a sample of 55 mice, a biologist found that 44% were able to run a maze in 30 seconds or less. Find the 99% limit for the population proportion of mice who can run a maze in 30 seconds or less.

Answer 1

The 99% confidence interval for the population proportion of mice who can run a maze in 30 seconds or less is
`\( \bar{p} \pm Z_(\alpha/2) \sqrt{\frac{\bar{p}(1-\bar{p})}{n}} \)`. Given that
`\(\bar{p} = 0.44\), \(n = 55\)`, and for a 99% confidence interval,
`\(Z_(\alpha/2) \approx 2.576\)`, the calculation yields an interval of approximately
`\(0.44 \pm 0.126\)`, resulting in a range of
`\( (0.314, 0.566) \).` Therefore, we can be 99% confident that the true proportion of mice able to run a maze in 30 seconds or less falls within this interval.

Step-by-step explanation:

To determine the 99% confidence interval for the population proportion
`(\( \bar{p} \))`, we use the formula
`\( \bar{p} \pm Z_(\alpha/2) \sqrt{\frac{\bar{p}(1-\bar{p})}{n}} \), where \(Z_(\alpha/2)\)` is the critical value for the desired confidence level,
`\(n\)` is the sample size, and
`\(\bar{p}\)` is the sample proportion. In this case, with a sample size of
`\(n = 55\)` and a sample proportion of
`\( \bar{p} = 0.44 \)`, and using the critical value
`\(Z_(\alpha/2) \approx 2.576\)` for a 99% confidence interval, the calculation produces an interval of
`\(0.44 \pm 0.126\).`

Interpreting the results, we can express with 99% confidence that the true population proportion of mice capable of running a maze in 30 seconds or less lies within the range of
`\( (0.314, 0.566) \)`. This interval provides a measure of the precision of our estimate and reflects the variability inherent in sampling.

The confidence interval approach allows us to make a statement about the likely range of the true population proportion based on the information obtained from the sample.

Answer 2

In this case, the 99% confidence interval for the population proportion of mice who can run a maze in 30 seconds or less is approximately 0.2823 to 0.5977.

Step-by-step explanation:

To find the 99% limit for the population proportion of mice who can run a maze in 30 seconds or less, we can use the formula for calculating confidence intervals.

1. Calculate the standard error: The standard error measures the variability in sample proportions. It can be calculated using the formula:

SE = √((p * (1 - p)) / n)

where p is the sample proportion and n is the sample size.

In this case, the sample proportion is 0.44 (44%) and the sample size is 55.

SE = √((0.44 * (1 - 0.44)) / 55) ≈ 0.0612

2. Calculate the margin of error: The margin of error determines the range within which the population proportion is likely to fall. It can be calculated by multiplying the standard error by the critical value from the standard normal distribution corresponding to the desired level of confidence.

For a 99% confidence level, the critical value is approximately 2.576.

Margin of error = critical value * standard error = 2.576 * 0.0612 ≈ 0.1577

3. Calculate the lower and upper limits of the confidence interval: Subtract and add the margin of error from the sample proportion to obtain the lower and upper limits of the confidence interval, respectively.

• Lower limit = sample proportion - margin of error = 0.44 - 0.1577 ≈ 0.2823
• Upper limit = sample proportion + margin of error = 0.44 + 0.1577 ≈ 0.5977

4. Interpretation: The 99% confidence interval for the population proportion of mice who can run a maze in 30 seconds or less is approximately 0.2823 to 0.5977. This means that we are 99% confident that the true proportion of mice who can run the maze in 30 seconds or less lies within this range.