Questions: The number of chocolate chips in an 18-ounce bag of chocolate chip cookies is approximately normally distributed with a (a) What is the probability that a randomly selected bag contains between 1000 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 1175 chocolate chips? (d) What is the percentile rank of a bag that contains 1000 chocolate chips? (a) The probability that a randomly selected bag contains between 1000 and 1400 chocolate chips, inclusive, is 0.8490 . (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 1175 chocolate chips is 0.7247 . (Round to four decimal places as needed.) (d) A bag that contains 1000 chocolate chips is in the rd percentile. (Round to the nearest integer as needed.)

The number of chocolate chips in an 18-ounce bag of chocolate chip cookies is approximately normally distributed with a (a) What is the probability that a randomly selected bag contains between 1000 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 1175 chocolate chips? (d) What is the percentile rank of a bag that contains 1000 chocolate chips? (a) The probability that a randomly selected bag contains between 1000 and 1400 chocolate chips, inclusive, is 0.8490 . (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 1175 chocolate chips is 0.7247 . (Round to four decimal places as needed.) (d) A bag that contains 1000 chocolate chips is in the rd percentile. (Round to the nearest integer as needed.)
Transcript text: The number of chocolate chips in an 18-ounce bag of chocolate chip cookies is approximately normally distributed with a (a) What is the probability that a randomly selected bag contains between 1000 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 1175 chocolate chips? (d) What is the percentile rank of a bag that contains 1000 chocolate chips? (a) The probability that a randomly selected bag contains between 1000 and 1400 chocolate chips, inclusive, is 0.8490 . (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 1175 chocolate chips is 0.7247 . (Round to four decimal places as needed.) (d) A bag that contains 1000 chocolate chips is in the rd percentile. (Round to the nearest integer as needed.)
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Solution

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Solution Steps

Step 1: Probability Between 1000 and 1400 Chips

To find the probability that a randomly selected bag contains between 1000 and 1400 chocolate chips, we calculate:

\[ P(1000 \leq X \leq 1400) = \Phi(Z_{end}) - \Phi(Z_{start}) \]

Where:

  • \( Z_{end} = \frac{1400 - 1260}{118} \approx 1.1864 \)
  • \( Z_{start} = \frac{1000 - 1260}{118} \approx -2.2034 \)

Thus, we have:

\[ P(1000 \leq X \leq 1400) = \Phi(1.1864) - \Phi(-2.2034) \approx 0.8685 \]

Step 2: Probability Fewer Than 1000 Chips

Next, we calculate the probability that a randomly selected bag contains fewer than 1000 chocolate chips:

\[ P(X < 1000) = \Phi(Z_{start}) - \Phi(-\infty) \]

Where:

  • \( Z_{start} = -2.2034 \)

Thus, we have:

\[ P(X < 1000) = \Phi(-2.2034) - 0 \approx 0.0138 \]

Step 3: Proportion More Than 1175 Chips

Finally, we find the proportion of bags that contain more than 1175 chocolate chips:

\[ P(X > 1175) = \Phi(\infty) - \Phi(Z_{start}) \]

Where:

  • \( Z_{start} = \frac{1175 - 1260}{118} \approx -0.7203 \)

Thus, we have:

\[ P(X > 1175) = 1 - \Phi(-0.7203) \approx 0.7643 \]

Final Answer

  • (a) Probability between 1000 and 1400 chips: \( \boxed{0.8685} \)
  • (b) Probability fewer than 1000 chips: \( \boxed{0.0138} \)
  • (c) Proportion more than 1175 chips: \( \boxed{0.7643} \)
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