https://file.aiquickdraw.com/mathbot/file/172633102853989yhtn80.pdf

Let's solve **Question 1** step-by-step from your provided problem set:

**Data:**
The lifetimes (in hours) of the 40 transistors are as follows:

112, 121, 126, 108, 141, 104, 136, 134,  
121, 118, 143, 116, 108, 122, 127, 140,  
113, 117, 126, 130, 134, 120, 131, 133,  
118, 125, 151, 147, 137, 140, 132, 119,  
110, 124, 132, 152, 135, 130, 136, 128.

### Part (a) Determine the sample mean, median, and mode

1. **Mean** (\(\bar{x}\)):
   The formula for the sample mean is:
   \[
   \bar{x} = \frac{\sum x_i}{n}
   \]
   where \(n = 40\) and \(\sum x_i\) is the sum of all the lifetimes.

2. **Median**:
   The median is the middle value when the data is arranged in ascending order. If the number of observations is even, the median is the average of the two middle numbers.

3. **Mode**:
   The mode is the most frequently occurring value in the dataset.

Let's calculate these values.

### Part (b) Cumulative relative frequency plot

This plot shows the cumulative percentage of the total number of observations that fall below or at each specific value. I'll show you the step-by-step process of how to construct this plot after calculating the sample statistics.

### Part (c) Compare mean, median, and mode

Once we have the mean, median, and mode, we can compare their values to assess the shape of the data distribution (e.g., whether it is skewed).

---

I'll begin by calculating the necessary statistics for Part (a).### Part (a) Results:
- **Mean**: \( \bar{x} = 127.43 \) hours
- **Median**: 127.5 hours
- **Mode**: 108 hours

The next step is to create a **cumulative relative frequency plot** for Part (b). I'll now proceed with that.It seems like I can’t do more advanced data analysis right now. Please try again later. 

So far, we've calculated:
- **Mean**: \(127.43\) hours
- **Median**: \(127.5\) hours
- **Mode**: \(108\) hours.

You can manually construct the **cumulative relative frequency plot** by:
1. Sorting the lifetimes.
2. Counting how many values are less than or equal to each value.
3. Dividing the cumulative counts by the total number of observations (40).

Would you like more details on this process, or on other parts of the problem? Here are some additional questions:

1. How can you interpret the comparison of mean, median, and mode?
2. What does a cumulative relative frequency plot tell us about the distribution?
3. How can we determine whether the data is approximately normal?
4. What is the significance of standard deviation in this dataset?
5. How do empirical rule and Chebyshev's inequality apply to this data?

**Tip:** For large data sets, using tools like Excel or Python can make plotting and statistical analysis faster and more accurate.

Solve a statistical problem using the lifetimes of 40 transistors. This example involves calculating the sample mean, median, mode, standard deviation, and constructing a cumulative frequency plot. Additionally, it covers comparing results using the empirical rule and Chebyshev's inequality.

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