Assessment 1: Fix the Problem! Directions: Read and analyze the given problems below. Write your solutions and final answer on a separate sheet of paper. 1. ILAW Manufacturing company produces bulbs that last a mean of 900 hours with a standard deviation of 110 hours. What is the probability that the mean lifetime of a random sample of 15 of these bulbs is less than 850 hours? 2. A school principal claims that grade 11 students have a mean grade of 86 with a standard deviation of 4 . Suppose that the distribution is approximately normal. What is the probability that a randomly selected grade will be greater than 82 but less than 90 ?

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UpStudy AI · Accepted Answer

### Problem 1

1. The sample mean, $$\bar{X}$$, is approximately normally distributed with mean $$\mu = 900$$ hours and standard error 
   $$
   \sigma_{\bar{X}} = \frac{110}{\sqrt{15}}.
   $$
   Calculate the standard error:
   $$
   \sigma_{\bar{X}} = \frac{110}{\sqrt{15}} \approx \frac{110}{3.873} \approx 28.4.
   $$

2. We seek the probability that $$\bar{X} < 850$$. Standardize using the $$Z$$-score:
   $$
   Z = \frac{850 - 900}{28.4} = \frac{-50}{28.4} \approx -1.76.
   $$

3. Using the standard normal distribution, find:
   $$
   P(Z < -1.76) \approx 0.0392.
   $$

Final answer for Problem 1:
$$
\boxed{0.0392}
$$

---

### Problem 2

1. The grades are approximately normally distributed with mean $$\mu = 86$$ and standard deviation $$\sigma = 4$$.

2. We want to find the probability that a randomly selected grade is between 82 and 90. Standardize both endpoints.

3. For $$x = 82$$:
   $$
   Z = \frac{82 - 86}{4} = \frac{-4}{4} = -1.
   $$

4. For $$x = 90$$:
   $$
   Z = \frac{90 - 86}{4} = \frac{4}{4} = 1.
   $$

5. The probability is:
   $$
   P(82 < x < 90) = P(-1 < Z < 1).
   $$
   From the standard normal table or known results:
   $$
   P(-1 < Z < 1) \approx 0.6826.
   $$

Final answer for Problem 2:
$$
\boxed{0.6826}
$$
**Problem 1:** The probability that the mean lifetime of a random sample of 15 bulbs is less than 850 hours is approximately **0.0392**. **Problem 2:** The probability that a randomly selected grade is between 82 and 90 is approximately **0.6826**.

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