. Over the last 15 Major League Baseball seasons, the mean number of strikeouts by the American League leader is 258.5. Assuming that the number of strikeouts by the league leader is normally distributed and the standard deviation for all seasons in all leagues is 34.9, find the 82% Confidence Interval and the 93% Confidence Interval for the mean.
I assume that you want the range of maximum strikeouts, centered about the mean, that the number falls within 82% or 93% of the time.
Using the normal distribution function, I get 93% probability for being between 196 and 323, and 82% probability for being between 215.5 and 301.5.
I used http://psych.colorado.edu/~mcclella/java/normal/accurateNormal.html
To find the confidence intervals for the mean number of strikeouts, we will use the formula:
CI = X̄ ± Z * (σ/√n)
where CI is the confidence interval, X̄ is the mean number of strikeouts, Z is the standard score corresponding to the desired confidence level, σ is the standard deviation, and n is the sample size.
Let's calculate the confidence intervals step by step:
1. 82% Confidence Interval:
- Calculate the critical value for an 82% confidence level. Since the distribution is symmetric, we can split the remaining probability equally on both tails, giving us (100% - 82%) / 2 = 9% on each tail. The critical value for a 9% tail is 0.09. We can look up this value in the standard normal distribution table or use a statistical calculator, and it corresponds to approximately 1.43.
- Substitute the values into the formula:
CI = 258.5 ± 1.43 * (34.9/√15)
- Calculate the standard error (SE) by dividing the standard deviation (σ) by the square root of the sample size (√n):
SE = 34.9/√15
- Calculate the confidence interval:
CI = 258.5 ± 1.43 * SE
2. 93% Confidence Interval:
- Calculate the critical value for a 93% confidence level. Similar to the previous calculation, we have (100% - 93%) / 2 = 3.5% on each tail. The critical value for a 3.5% tail is 0.035. Looking this up in the standard normal distribution table or using a statistical calculator, we find it corresponds to approximately 1.81.
- Substitute the values into the formula:
CI = 258.5 ± 1.81 * (34.9/√15)
- Calculate the standard error (SE):
SE = 34.9/√15
- Calculate the confidence interval:
CI = 258.5 ± 1.81 * SE
By substituting the values and performing the calculations, we can find the confidence intervals for the mean number of strikeouts for both the 82% and 93% confidence levels.