To solve this problem, follow these detailed steps:
1. Identify the Sample Data:
The sample data provided consists of 40 units with the following lifespans (in days):
```
[39, 31, 38, 40, 29, 32, 33, 39, 35, 32, 32, 27, 30, 31, 27,
30, 29, 34, 36, 25, 30, 32, 38, 35, 40, 29, 32, 31, 26, 26,
32, 26, 30, 40, 32, 39, 37, 25, 29, 34]
```
2. Calculate the Point Estimate of the Population Mean:
To find the point estimate of the population mean, you compute the mean (average) of the sample data:
[tex]\[
\text{Population Mean Estimate} = \text{Mean} = \frac{\sum{x}}{n}
\][/tex]
where [tex]\( x \)[/tex] is each individual lifespan value, and [tex]\( n \)[/tex] is the number of samples (40 in this case).
After performing the calculation, the point estimate of the population mean is:
[tex]\[
32.3
\][/tex]
3. Determine the Threshold for Defectiveness:
A product is considered defective if its lifespan is less than 26 days.
4. Calculate the Proportion of Defective Units:
To calculate the proportion of defective units, count how many units have a lifespan of less than 26 days and divide it by the total number of units (40):
[tex]\[
\text{Proportion of Defective Units} = \frac{\text{Number of Defective Units}}{n}
\][/tex]
After counting, there are 2 units with a lifespan of less than 26 days out of the 40 units.
The proportion of defective units is:
[tex]\[
0.05
\][/tex]
Thus, the point estimate of the population mean is [tex]\( 32.3 \)[/tex] and the point estimate of the proportion of defective units is [tex]\( 0.05 \)[/tex].
So, the final answer is:
The point estimate of the population mean is [tex]\( 32.3 \)[/tex] and the point estimate of the proportion of defective units is [tex]\( 0.05 \)[/tex].
