Answer :
To find the point estimates, we need to calculate two things:
1. The point estimate of the population mean.
2. The point estimate of the proportion of defective units.
### Calculation of the Point Estimate of the Population Mean:
The point estimate of the population mean is the average lifespan of the sample data. The sample data provided is:
[tex]\[ 39, 31, 38, 40, 29, 32, 33, 39, 35, 32, 32, 27, 30, 31, 27, 30, 29, 34, 36, 25, 30, 32, 38, 35, 40, 29, 32, 31, 26, 26, 32, 26, 30, 40, 32, 39, 37, 25, 29, 34 \][/tex]
The point estimate of the population mean is calculated as:
[tex]\[ \text{Point estimate of the population mean} = \frac{\sum \text{sample data}}{\text{number of samples}} \][/tex]
Summing all the sample data, we get 1292. Dividing by the number of samples (40), we get:
[tex]\[ \frac{1292}{40} = 32.3 \][/tex]
### Calculation of the Point Estimate of the Proportion of Defective Units:
A product is considered defective if its lifespan is less than 26 days. We count the number of defective units in the sample data and then divide that by the total number of units in the sample.
Counting the defective units (those less than 26 days), we find that there are 2 defective units.
The point estimate of the proportion of defective units is:
[tex]\[ \text{Point estimate of the proportion of defective units} = \frac{\text{number of defective units}}{\text{total number of units}} \][/tex]
So,
[tex]\[ \frac{2}{40} = 0.05 \][/tex]
### Final Answers:
The point estimate of the population mean is [tex]\( \boxed{32.3} \)[/tex] and the point estimate of the proportion of defective units is [tex]\( \boxed{0.05} \)[/tex].
1. The point estimate of the population mean.
2. The point estimate of the proportion of defective units.
### Calculation of the Point Estimate of the Population Mean:
The point estimate of the population mean is the average lifespan of the sample data. The sample data provided is:
[tex]\[ 39, 31, 38, 40, 29, 32, 33, 39, 35, 32, 32, 27, 30, 31, 27, 30, 29, 34, 36, 25, 30, 32, 38, 35, 40, 29, 32, 31, 26, 26, 32, 26, 30, 40, 32, 39, 37, 25, 29, 34 \][/tex]
The point estimate of the population mean is calculated as:
[tex]\[ \text{Point estimate of the population mean} = \frac{\sum \text{sample data}}{\text{number of samples}} \][/tex]
Summing all the sample data, we get 1292. Dividing by the number of samples (40), we get:
[tex]\[ \frac{1292}{40} = 32.3 \][/tex]
### Calculation of the Point Estimate of the Proportion of Defective Units:
A product is considered defective if its lifespan is less than 26 days. We count the number of defective units in the sample data and then divide that by the total number of units in the sample.
Counting the defective units (those less than 26 days), we find that there are 2 defective units.
The point estimate of the proportion of defective units is:
[tex]\[ \text{Point estimate of the proportion of defective units} = \frac{\text{number of defective units}}{\text{total number of units}} \][/tex]
So,
[tex]\[ \frac{2}{40} = 0.05 \][/tex]
### Final Answers:
The point estimate of the population mean is [tex]\( \boxed{32.3} \)[/tex] and the point estimate of the proportion of defective units is [tex]\( \boxed{0.05} \)[/tex].
