When Brody goes bowling, his scores are normally distributed with a mean of 125 and a standard deviation of 11. What is the probability that the next game Brody bowls, his score will be between 97 and 115, to the nearest thousandth?
To find the probability that Brody's score will be between 97 and 115, we need to calculate the z-scores for each of these scores and then use the standard normal distribution to find the probabilities.
First, calculate the z-scores:
For a score of 97:
z = (97 - 125) / 11 = -2.545
For a score of 115:
z = (115 - 125) / 11 = -0.909
Next, we need to find the probabilities associated with these z-scores using a standard normal distribution table or calculator. The probability that Brody's score will be between 97 and 115 is equal to the sum of the probabilities to the left of each z-score.
P(97 < X < 115) = P(X < 115) - P(X < 97)
P(97 < X < 115) = P(Z < -0.909) - P(Z < -2.545)
P(97 < X < 115) = 0.1838 - 0.0054
P(97 < X < 115) = 0.1784
Therefore, the probability that Brody's score will be between 97 and 115 is approximately 0.178, or 17.8% to the nearest thousandth.
is this right?
I made a mistake with the calculations, let's correct this:
To find the probability that Brody's score will be between 97 and 115, we need to calculate the z-scores for each of these scores and then use the standard normal distribution to find the probabilities.
Calculate the z-score for a score of 97:
z = (97 - 125) / 11 = -28 / 11 = -2.5455
Calculate the z-score for a score of 115:
z = (115 - 125) / 11 = -10 / 11 = -0.9091
Next, we need to find the probabilities associated with these z-scores using a standard normal distribution table or calculator. The probability that Brody's score will be between 97 and 115 is equal to the area under the curve between the two z-scores.
Using a standard normal distribution table or calculator, we find:
P(-2.5455 < Z < -0.9091) = P(Z < -0.9091) - P(Z < -2.5455)
P(-2.5455 < Z < -0.9091) = 0.1821 - 0.0054
P(-2.5455 < Z < -0.9091) = 0.1767
Therefore, the corrected probability that Brody's score will be between 97 and 115 is approximately 0.1767, or 17.67% to the nearest thousandth. Thank you for pointing out the error, and I appreciate your attention to detail.