An athletic apparel manufacturer randomly selects 200 high school varsity basketball players. A mean height of 6.18 feet is calculated with a margin of error of ±0.15 for a 99% confidence interval.
To calculate the 99% confidence interval for the mean height, we need to use the formula:
Confidence Interval = mean ± (critical value × standard error)
The critical value is taken from the z-table for a 99% confidence level, which is 2.576.
The standard error can be calculated using the formula:
Standard Error = standard deviation / sqrt(sample size)
Since the sample size is 200, let's assume the standard deviation is known to be 0.5 feet.
Plugging these values into the formula:
Standard Error = 0.5 / sqrt(200) ≈ 0.035355
Now we can calculate the confidence interval:
Confidence Interval = 6.18 ± (2.576 × 0.035355)
Confidence Interval = 6.18 ± 0.091
Therefore, the 99% confidence interval for the mean height of high school varsity basketball players is approximately 6.09 to 6.27 feet.
Suppose you survey 500 shoppers at the Mall of America. A parameter of this population is
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Unfortunately, without knowing what the parameter it is that we are interested in (e.g., average age, proportion of shoppers who own a car, etc.), it is not possible to provide a specific calculation or an exact confidence interval. Can you please provide more details about the parameter of interest?
To calculate a confidence interval for the mean height of the high school varsity basketball players, we can use the following formula:
Confidence Interval = Mean ± (Z * Standard Error)
Where:
- Confidence Interval is the range within which we can be confident that the true population mean lies.
- Mean is the sample mean, which is 6.18 feet in this case.
- Z is the Z-score corresponding to the desired level of confidence. For a 99% confidence interval, the Z-score is approximately 2.576.
- Standard Error is the standard deviation of the sample divided by the square root of the sample size.
To find the standard error, we need to know the standard deviation of the sample. Without that information, we won't be able to calculate the confidence interval.
To calculate the mean height of a sample of high school varsity basketball players with a 99% confidence interval and a margin of error, follow these steps:
1. Determine the formula for the confidence interval:
The formula to calculate the confidence interval for a population mean is:
CI = X̄ ± Z * (σ / √n)
Where:
- CI is the confidence interval
- X̄ is the sample mean
- Z is the Z-score corresponding to the desired level of confidence
- σ is the population standard deviation (unknown in this case)
- n is the sample size
2. Find the Z-score for a 99% confidence level:
The Z-score corresponding to a 99% confidence level is approximately 2.576. This value represents the number of standard deviations from the mean that corresponds to a 99% confidence level.
3. Substitute the given values into the formula:
We are given that the mean height (X̄) is 6.18 feet, and the margin of error is ±0.15. Since the margin of error gives a range around the mean, we can divide it by 2 to determine the value of σ / √n.
Given: X̄ = 6.18 feet and Margin of Error (E) = ±0.15 feet
σ / √n = E / 2
Solve for σ / √n:
σ / √n = 0.15 / 2 = 0.075
Now we can substitute the values into the confidence interval formula:
CI = X̄ ± Z * (σ / √n)
CI = 6.18 ± 2.576 * 0.075
4. Calculate the confidence interval:
To calculate the upper and lower limits of the confidence interval, we just need to substitute the values:
Upper Limit = X̄ + Z * (σ / √n)
Upper Limit = 6.18 + 2.576 * 0.075
Lower Limit = X̄ - Z * (σ / √n)
Lower Limit = 6.18 - 2.576 * 0.075
Calculate the upper and lower limits, rounding to the desired precision.
By following these steps, you can calculate the 99% confidence interval for the mean height of high school varsity basketball players, based on the given sample of 200 players.