The problem in the image is related to statistics, specifically on estimating the mean, standard deviation, and testing the accuracy of a claim about a machine part's lifespan.

Let's break it down:

Given Data:

The table represents lifetimes of machine parts and their frequencies:

Total frequency: 50 (number of parts tested).

Manufacturer's Claim:

• Claim: 90% of the parts lasted longer than one standard deviation below the mean.

Task:

We need to:

1. Estimate the mean of the lifetimes.
2. Estimate the standard deviation of the lifetimes.
3. Test the claim: Determine how many parts lasted longer than one standard deviation below the mean.

1. Mean Estimation:

To estimate the mean, we first calculate the midpoint for each group:

• Midpoint for $5 < h \leq 10$ = $\frac{5 + 10}{2} = 7.5$
• Midpoint for $10 < h \leq 15$ = $\frac{10 + 15}{2} = 12.5$
• Midpoint for $15 < h \leq 20$ = $\frac{15 + 20}{2} = 17.5$
• Midpoint for $20 < h \leq 25$ = $\frac{20 + 25}{2} = 22.5$
• Midpoint for $25 < h \leq 30$ = $\frac{25 + 30}{2} = 27.5$

Next, we calculate the estimated mean $\mu$ using the formula: $\mu = \frac{\sum (f_i \times x_i)}{\sum f_i}$ where $f_i$ is the frequency and $x_i$ is the midpoint.

2. Standard Deviation Estimation:

For standard deviation $\sigma$, we use the formula: $\sigma = \sqrt{\frac{\sum f_i (x_i - \mu)^2}{\sum f_i}}$ where $x_i$ is the midpoint for each class and $\mu$ is the mean.

3. Test the Claim:

Once we have the mean and standard deviation, we compute the value of: $\mu - \sigma$ We then check how many parts lasted longer than this value and compare it to the claim that 90% lasted longer than $\mu - \sigma$.

Would you like me to proceed with detailed calculations for the mean and standard deviation? If so, I can guide you through each step.

Follow-up Questions:

1. What is the significance of calculating the mean in this context?
2. How do the midpoints help in calculating an estimated mean?
3. Why is the standard deviation crucial for testing the manufacturer's claim?
4. How would the result change if the frequency distribution was different?
5. Could this data be represented using another method like a histogram, and how would that help in understanding the wear-out times?

Tip: When calculating standard deviation from grouped data, remember that it's only an estimate since the midpoints are used to approximate the data points.