First, let's determine the point estimate of the population mean from the sample data provided.
To find the population mean estimate, we calculate the average of the sample data. Summing the sample data and dividing by the number of units in the sample:
The sum of the sample values:
[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 number of units in the sample:
[tex]\[ 40 \][/tex]
Thus, the population mean estimate (average lifespan) is:
[tex]\[ \frac{\text{sum of the sample values}}{\text{number of units}} \][/tex]
From the calculations, the population mean estimate is [tex]\( 32.3 \)[/tex].
Next, we determine the point estimate of the proportion of defective units. A unit is considered defective if its lifespan is less than 28 days. We need to count the number of units in the sample that meet this criterion:
Defective units:
[tex]\[ 27, 27, 25, 25, 26, 26, 26, 25 \][/tex]
This gives us 7 defective units.
The proportion of defective units is thus:
[tex]\[ \frac{\text{number of defective units}}{\text{number of units}} \][/tex]
[tex]\[ \frac{7}{40} \][/tex]
From the calculations, the proportion of defective units is [tex]\( 0.175 \)[/tex].
So, the point estimate of the population mean is [tex]\( 32.3 \)[/tex], and the point estimate of the proportion of defective units is [tex]\( 0.175 \)[/tex].
