The number of chocolate chips in an 18-ounce bag of chocolate chip cookies is approximately normally distributed with a mean of 1252 chips and standard deviation 129 chips.
(a) What is the probability thaf a randomly selected bag contains between 1100 and 1400 chocolate chips, inclusive?
(b) What is the probability that a randomly selected bag contains fewer than 1000 chocolate chips?
(c) What proportion of bags contains more than 1225 chocolate chips?
(d) What is the percentile rank of a bag that contains 1425 chocolate chips?
(a) The probability that a randomly selected bag contains between 1100 and 1400 chocolate chips, inclusive, is 0.7550 .
(Round to four decimal places as needed.)
(b) The probability that a randomly selected bag contains fewer than 1000 chocolate chips is 0.0254 .
(Round to four decimal places as needed.)
(c) The proportion of bags that contains more than 1225 chocolate chips is (Round to four decimal places as needed.)
Answer
Final Answer: The probability that a randomly selected bag contains between 1100 and 1400 chocolate chips, inclusive, is approximately \(\boxed{0.7550}\).
Step 1 ：We are given a normal distribution with a mean of 1252 and a standard deviation of 129. We are asked to find the probability that a randomly selected bag contains between 1100 and 1400 chocolate chips. This is equivalent to finding the area under the normal distribution curve between these two values.
Step 2 ：To do this, we first need to convert these values to z-scores, which give us the number of standard deviations away from the mean a value is. The formula for a z-score is: \(z = \frac{X - \mu}{\sigma}\) where X is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
Step 3 ：Using the z-score formula, we find that the z-score for 1100 is approximately -1.178 and the z-score for 1400 is approximately 1.147.
Step 4 ：We can then use a z-table to find the probabilities associated with these z-scores. The probability associated with the z-score for 1100 is approximately 0.119 and the probability associated with the z-score for 1400 is approximately 0.874.
Step 5 ：The probability that a randomly selected bag contains between 1100 and 1400 chocolate chips is the probability associated with the z-score for 1400 minus the probability associated with the z-score for 1100. This gives us a probability of approximately 0.755.
Step 6 ：Final Answer: The probability that a randomly selected bag contains between 1100 and 1400 chocolate chips, inclusive, is approximately \(\boxed{0.7550}\).
