W - 7.2
Question 15, 7.2.39
HW Score: $84.44 \%, 12.67$ of 15 points
Part 1 of 4
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The number of chocolate chips in an 18-ounce bag of chocolate chip cookies is approximately normally distributed with mean 1252 and standard deviation 129 chips.
(a) What is the probability that a randomly selected bag contains between 1000 and 1500 chocolate chips?
(b) What is the probability that a randomly selected bag contains fewer than 1125 chocolate chips?
(c) What proportion of bags contains more than 1200 chocolate chips?
(d) What is the percentile rank of a bag that contains 1000 chocolate chips?
Click the icon to view a table of areas under the normal curve
(a) The probability that a randomly selected bag contains between 1000 and 1500 chocolate chips is $\square$.
(Round to four decimal places as needed)
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Final Answer: The probability that a randomly selected bag contains between 1000 and 1500 chocolate chips is \( \boxed{0.9473} \).
Step 1 ：First, we need to calculate the z-scores for 1000 and 1500. The mean (μ) is 1252 and the standard deviation (σ) is 129. The formula for a z-score is \( (X - μ) / σ \). So, the z-score for 1000 is \( (1000 - 1252) / 129 \) and the z-score for 1500 is \( (1500 - 1252) / 129 \).
Step 2 ：Then, we need to find the probabilities associated with these z-scores. The probability of a bag containing between 1000 and 1500 chips is the difference between the probabilities of the bag containing less than 1500 chips and the bag containing less than 1000 chips.
Step 3 ：Let's calculate these values. The z-score for 1000 is -1.9534883720930232 and the z-score for 1500 is 1.9224806201550388. The probability for 1000 is 0.025380876893736335 and the probability for 1500 is 0.9727273450797767.
Step 4 ：The probability of a bag containing between 1000 and 1500 chips is the difference between the probabilities of the bag containing less than 1500 chips and the bag containing less than 1000 chips, which is \( 0.9727273450797767 - 0.025380876893736335 = 0.9473464681860403 \).
Step 5 ：The probability that a randomly selected bag contains between 1000 and 1500 chocolate chips is approximately 0.9473, or 94.73%. This means that about 94.73% of bags will contain between 1000 and 1500 chips.
Step 6 ：Final Answer: The probability that a randomly selected bag contains between 1000 and 1500 chocolate chips is \( \boxed{0.9473} \).
