Question

If a baseball player's batting average is 0.340 or 34%, find the probability that the player will have a bad season and only score at most 60 hits in 200 times at bat?

Answer 1

X ~ Binomial(n-200, p-0.34) Binomial can be approximated to normal with np 200* 0.34 -68 σーyMp(1-p)-V 200 (0.34) (1 0.34) 6.699254 6.699 P(X 60) Since we are approximating a discrete distribution by continuous normal dis- tribution, values between 59.5 and 60.5 both approximate to 60. Thus, "less than or equal to 60 " corresponds to continuous normal distribution with X < 60.5 after continuity correction Normal Distribution, X-605, 1-68, ơ-6.699 We convert this to standard normal usingz - 60.5 68 6.699 -1.119528 -1.12 P(X < 60.5) Area to the left of 60.5 68 σ 6.699254 P(X < 60.5 60.5 68 P(X < 60.5) P(Z -1.12)0.1314 (from z-table) P(X 3 60) 0.1314 If excel is used instead of z-table, it would be: 0.1314575537 Note that if this question is solved without continuity correction, the answer would have been P(X < 60) P(Z <-1.19) = 0.1170 Without integer-continuity correction:P(X60) 0.1170 If excel is used instead of z-table, it would be: 0.1162071433

Answer 2

Binomial Problem with n= 200 ; p = 0.34 ; x =60

By using normal approximation,
X~Normal(μ=n*p=200*0.34=68,σ=√n*p*(1-p)=√200*0.34*(1-0.34)=6.7)

So the probability is
P(X<60)
=P((X-μ)/σ < (60-68)/6.7)
=P(Z< -1.19)
=0.117 (check normal table)

Answer 3

sample size, n= 200
proportion,p = 0.34

mean, μ=n*p=200*0.34=68,
standard deviation,σ=√n*p*(1-p)=√200*0.34*(1-0.34)=6.7

So the probability the player will have a bad season and only scoreat most 60 hits in 200 times at bat

P(X60)
=P()
=P( z -1.19)
=0.117                            (from z table)

Answer 4

Step 1: Define terms.
p = population proportion =0.34
p_h = sample proportion (or p hat) = 60/200 = 0.30
n = sample size = 200
Step 2: Find the standard error.
SE(p_h) =standard error of p hat = √[ p * (1 – p) / n]
   = √(0.34 * 0.66 / 200) = 0.03350
Step 3: Find the z-value.
z = (p_h – p) /SE(p_h)
    = ( 0.30 – 0.34) / 0. 03350 = -1.19
Step 4: Find the probability.
P(p_h < 0.30)= P(Z < -1.91) =0.1162

Hope this helps,

Mike