a magazine reports that the average bowling score for league bowlers in the united States is 157 with standard deviation 12, scores are approximately normally distributed.
what's the probability a random sample of 150 bowlers will have an average score greater than 165.
mean= 157 sd= 0.98
I worked it out by finding the z-score but the z score ended up being 8.16 which is impossible, i'm so confused please help!

Question

a magazine reports that the average bowling score for league bowlers in the united States is 157 with standard deviation 12, scores are approximately normally distributed.
what's the probability a random sample of 150 bowlers will have an average score greater than 165.
mean= 157 sd= 0.98
I worked it out by finding the z-score but the z score ended up being 8.16 which is impossible, i'm so confused please help!

fibonaccichick666 · Answer

For your reading pleasure because I agree with your z-score http://www.oswego.edu/~srp/158/Z%20Scores/Z%20Scores.pdf

kropot72 · Answer

You need to consider that "sd = 0.98" means the standard deviation of X-bar. It is sometimes called the 'standard error'. In this case the standard error has been calculated from:
$$\large S.E.=\frac{\sigma}{\sqrt{n}}=\frac{12}{\sqrt{150}}=0.98$$

anonymous · Answer

right, and then when i plugged that with the mean and x value into the z score formula i got 8.1 which i don't know what to do with that number :/ unless the sd is wrong i have no idea what to do @kropot72

fibonaccichick666 · Answer

8.1 is not impossible, just improbable

anonymous · Answer

But then what would be the probability over 100? @FibonacciChick666

fibonaccichick666 · Answer

if probability is 100%, that would mean that something 8 standard deviations from the mean is completely probable.  Is that true?

anonymous · Answer

probably not true most likely @FibonacciChick666

fibonaccichick666 · Answer

right, so as you get farther and farther away, the probability gets smaller and smaller.  What's the smallest it can be?

anonymous · Answer

0% probability right? @FibonacciChick666

fibonaccichick666 · Answer

right, but since we can't really have zero, we are rounding.  So we say approximately 0 chance

fibonaccichick666 · Answer

so, do you reject or fail to reject the null?

anonymous · Answer

I reject!! haha I don't really know what those mean @FibonacciChick666

fibonaccichick666 · Answer

oh, sorry, I teach that next week and was getting ahead of myself haha

anonymous · Answer

haha no problem i guess ill just answer about 0 probability cause the numbers don't lie even though it seems super wrong because for 150 bowlers not to have an average score greater than 165 doesn't make sense to me but whatever :) thank you for your help! @FibonacciChick666

fibonaccichick666 · Answer

yea, I don't get the semantics of it, but those are the numbers(so long as everything was copied correctly)!  Sorry I can't explain it better!  Best of luck!

anonymous · Answer

no problem and thanks @FibonacciChick666

kropot72 · Answer

Notice that the sample size is large (150), and the sample standard deviation is small (0.98). The mean of the samples is the same as the population mean. Therefore I think it makes perfect sense that the chance that a random sample of 150 bowlers will have an average score of 165 is negligibly small.