I'm having some difficulty with this problem:
Over the last 15 Major League Baseball seasons, the mean # of strikeouts by the American League leader is 258.5. Assuming that the # of strikeouts by the league leader is normally distributed & the standard deviation for all seasons in all leagues is 34.9, find the 82% Confidence Interval & the 93% Confidence Interval for the mean
Here's what I got for 82% CI:
x1=82%
μ=258.5
σ=34.9
n=15
α=.82
1- α=0.18
α/2=0.09
1-0.09=0.91
z-score=1.35
p^=x/n
p^=0.18/15
p^=0.988
q^=1-p^
q^=1-0.988
q^=0.012
E=zα/2*√p^q^/n
1.35*√(0.988)(0.012)/15
1.35*√0.011856/15
1.35*√7.904
1.35*2.811405343
E=3.795397212
p^ - E <p < p^ + E
0.988 – 3.795397212 < p < 0.988 + 3.795397212
-3.696597212 < p < 4.783397212
However, when I use computer software, I get 245.7828 < mean < 271.2172, which I think is correct. Of course, the software doesn't show me how to get to the answer.
If anyone can help figure out where I went wrong, please point me in the right direction.
Thanks!
Here's the formula I would use for this type of problem:
CI82 = mean + or - 1.35(sd / √n)
...where + or - 1.35 represents the 82% confidence interval, sd = standard deviation, √ = square root, and n = sample size.
With your data:
CI82 = 258.5 + or - 1.35 (34.9 / √15)
....... = 258.5 + or - 1.35 (9.01)
....... = 258.5 + or - 12.16
258.5 - 12.16 = 246.34
258.5 + 12.16 = 270.66
Interval is from 246.34 to 270.66. There may be differences due to rounding.
I hope this helps.
Thank you - that helped a lot!
I can help you understand where you went wrong in calculating the 82% Confidence Interval.
To find the 82% Confidence Interval, you will follow these steps:
Step 1: Determine the critical value
Since the data is normally distributed, you can use the Z-table or a calculator to find the critical value corresponding to an alpha level of 0.09. In this case, the critical value is approximately 1.35.
Step 2: Calculate the standard error (SE)
The standard error measures the average distance between each sample mean and the population mean. It is calculated by dividing the standard deviation by the square root of the sample size.
SE = σ/√n
SE = 34.9/√15
SE = 9.003
Step 3: Calculate the margin of error (ME)
The margin of error represents the range within which the population mean is likely to fall. It is calculated by multiplying the critical value by the standard error.
ME = Z * SE
ME = 1.35 * 9.003
ME = 12.154
Step 4: Calculate the Confidence Interval (CI)
To find the Confidence Interval, you subtract and add the margin of error from the sample mean.
CI = x̄ ± ME
CI = 258.5 ± 12.154
CI = 246.346 to 270.654
So, the correct 82% Confidence Interval for the mean number of strikeouts by the American League leader is 246.346 to 270.654.
I hope this clears up the confusion and helps you understand the correct calculation of the 82% Confidence Interval.