1. ## small samples urgent

I need help with Parts B and D of this question

John likes to play a game at the CNE where he throws balls at a target anda device measures how fast the balls are thrown. To win, he must throw the ball over 70 miles per hour 3 times in a row. Unfortunately, on average, john can only throw a ball 68 miles per hour, with a standard deviation of 8 (assume the distribution is normal).

a) what is the probability that John wins any time he plays?
b) what is the probability that john wins 12 times or more out of 100?
c) prior to last year's CNE, two things happened. first, the game was changed so that you only have to throw 2 balls in a row over 70 miles oer hour to win. Second, john did throwing exercises to improve his strength. The exercises seem to have worked, since John recently won 20 times in 30 tries. Use these results to test the hypothesis that he actually is stronger using alpha < 0.05.
d) Based on these 30 tries, construct a 99% confidence interval for the probability that John will win wny time he plays under the new rules.

a) 0.0646
b) 0.0202
c) Z = 7.54, reject Ho, John is significantly stronger
d) 0.4451 to 0.8883

__________________________________________________ ____________________________________
For Part B:

I assumed that a game consisted of getting 3 balls. To win, all must be thrown 70+ mph. John's probability of winning 1 game is .0646.

In 100 games we would expect him to win 6.46 games, with a standard deviation of sqrt(.0646*(1.-.0646)*100)=2.46

So the likelyhood of seeing 12 wins out of 100 is z=(12-6.46)/2.46 = 2.25, which corresponds to a .0122 but this is not the right answer

2. Originally Posted by skhan
I need help with Parts B and D of this question

John likes to play a game at the CNE where he throws balls at a target anda device measures how fast the balls are thrown. To win, he must throw the ball over 70 miles per hour 3 times in a row. Unfortunately, on average, john can only throw a ball 68 miles per hour, with a standard deviation of 8 (assume the distribution is normal).

a) what is the probability that John wins any time he plays?
b) what is the probability that john wins 12 times or more out of 100?
c) prior to last year's CNE, two things happened. first, the game was changed so that you only have to throw 2 balls in a row over 70 miles oer hour to win. Second, john did throwing exercises to improve his strength. The exercises seem to have worked, since John recently won 20 times in 30 tries. Use these results to test the hypothesis that he actually is stronger using alpha < 0.05.
d) Based on these 30 tries, construct a 99% confidence interval for the probability that John will win wny time he plays under the new rules.

a) 0.0646
b) 0.0202
c) Z = 7.54, reject Ho, John is significantly stronger
d) 0.4451 to 0.8883

__________________________________________________ ____________________________________
For Part B:

I assumed that a game consisted of getting 3 balls. To win, all must be thrown 70+ mph. John's probability of winning 1 game is .0646.

In 100 games we would expect him to win 6.46 games, with a standard deviation of sqrt(.0646*(1.-.0646)*100)=2.46

So the likelyhood of seeing 12 wins out of 100 is z=(12-6.46)/2.46 = 2.25, which corresponds to a .0122 but this is not the right answer
Because this is the normal approximation to a binomial the probability
of winning 12 times or more should be worked out using the z-score for
11.5 wins.

This is because in the normal approximation we calculate the probability
of exactly N wins as the area under the normal curve for between N-0.5
and N+0.5.

RonL

3. Originally Posted by skhan
I need help with Parts B and D of this question

John likes to play a game at the CNE where he throws balls at a target anda device measures how fast the balls are thrown. To win, he must throw the ball over 70 miles per hour 3 times in a row. Unfortunately, on average, john can only throw a ball 68 miles per hour, with a standard deviation of 8 (assume the distribution is normal).

a) what is the probability that John wins any time he plays?
b) what is the probability that john wins 12 times or more out of 100?
c) prior to last year's CNE, two things happened. first, the game was changed so that you only have to throw 2 balls in a row over 70 miles oer hour to win. Second, john did throwing exercises to improve his strength. The exercises seem to have worked, since John recently won 20 times in 30 tries. Use these results to test the hypothesis that he actually is stronger using alpha < 0.05.
d) Based on these 30 tries, construct a 99% confidence interval for the probability that John will win wny time he plays under the new rules.

a) 0.0646
b) 0.0202
c) Z = 7.54, reject Ho, John is significantly stronger
d) 0.4451 to 0.8883
For part d we are again going to use a normal approximation (in a rather
cavalier manner).

We have a point estimate of the probability that John wins of $\mu=0.6666..$,
and this has a standard deviation of:

$\sigma=(\sqrt{30\times (2/3)\times (1/3)})/30\approx 0.08607$

Now a $99 \%$ confidence interval extends to $\mu \pm 2.576 \sigma$, or: $\approx (0.44495,0.88838)$.

(the answer is quoted to more digits than the method warrants on the
basis of 30 trials, this number of trials probably warrants only two decimal
places).

RonL