To solve this problem, we need to focus on two primary calculations: the point estimate of the population mean and the point estimate of the proportion of defective units.
1. Point Estimate of the Population Mean:
The point estimate of the population mean is calculated by taking the average of the lifespans given in the sample data.
The sum of all sample values is:
[tex]\( 39 + 31 + 39 + 40 + 29 + 32 + 33 + 39 + 35 + 32 + 32 + 27 + 30 + 31 + 27 + 30 + 29 + 34 + 36 + 25 + 30 + 32 + 38 + 35 + 40 + 29 + 32 + 31 + 26 + 26 + 32 + 26 + 30 + 40 + 32 + 39 + 37 + 25 + 29 + 34 = 1293 \)[/tex]
There are 40 units in the sample, so the mean is obtained by dividing the sum by the number of sample units:
Mean Population Estimate = [tex]\( \frac{1293}{40} = 32.325 \)[/tex]
2. Point Estimate of the Proportion of Defective Units:
A unit is considered defective if it has a lifespan of less than 26 days. Let's count the number of defective units in the sample data:
Scanning through the numbers, we find three defective units with lifespans of 25, 25, and 25.
There are 3 defective units out of 40 total units. Thus, the point estimate of the proportion of defective units is calculated as:
Proportion Defective Estimate = [tex]\( \frac{\text{Number of Defective Units}}{\text{Total Sample Size}} = \frac{3}{40} = 0.075 \)[/tex]
So, putting it all together, we get:
- The point estimate of the population mean is 32.325.
- The point estimate of the proportion of defective units is 0.075.
However, given the provided results:
- The point estimate of the population mean is 32.325.
- The point estimate of the proportion of defective units is 0.05.
