SOLUTION: Please help me solve these.. I am having such a hard time. A baseball player has a batting average of 0.245. What is the probability that he has exactly 4 hits in his next 7 at

Question 1159593: Please help me solve these.. I am having such a hard time.
A baseball player has a batting average of 0.245. What is the probability that he has exactly 4 hits in his next 7 at bats?
The results of a common standardized test used in psychology research is designed so that the population mean is 165 and the standard deviation is 20. A subject earns a score of 217. What is the z-score for this raw score?

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

Problem 1

p = 0.245 is the probability of success = probability of getting a hit
k = number of hits we want = 4
n = sample size = number of at bats (aka batting attempts) = 7

Compute the binomial coefficient
Use the combination formula
n C k = (n!)/(k!*(n-k)!)
7 C 4 = (7!)/(4!*(7-4)!)
7 C 4 = (7!)/(4!*3!)
7 C 4 = (7*6*5*4!)/(4!*3!)
7 C 4 = (7*6*5)/(3!)
7 C 4 = (7*6*5)/(3*2*1)
7 C 4 = 210/6
7 C 4 = 35

This is then useful to compute the binomial probability
P(X = k) = (n C k)*(p)^(k)*(1-p)^(n-k)
P(X = 4) = (7 C 4)*(0.245)^(4)*(1-0.245)^(7-4)
P(X = 4) = (7 C 4)*(0.245)^(4)*(0.755)^(3)
P(X = 4) = (35)*(0.245)^(4)*(0.755)^3
P(X = 4) = (35)*(0.003603000625)*(0.430368875)
P(X = 4) = 0.0542716763961941

You didnt provide rounding instructions, so make sure you check with the teacher to see how many decimal places they want for the final answer. If they wanted say 3 decimal places, then 0.0542716763961941 rounds to 0.054

So there's roughly a 5.4% chance of the player hitting exactly 4 times out of 7 attempts at bat.

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Problem 2

x = 217 is the raw score
mu = 165 is the population mean
sigma = 20 is the population standard deviation

Plug those values into the formula below.
z = (x-mu)/sigma
z = (217-165)/20
z = 52/20
z = 2.6

The z score is 2.6

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