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A company manufactures 2,000 units of its flagship product in a day. The quality control department takes a random sample of 40 units to test for quality. The product is put through a wear-and-tear test to determine the number of days it can last. If the product has a lifespan of less than 28 days, it is considered defective. The table gives the sample data that a quality control manager collected.

\begin{tabular}{|l|l|l|l|l|}
\hline 39 & 31 & 38 & 40 & 29 \\
\hline 32 & 33 & 39 & 35 & 32 \\
\hline 32 & 27 & 30 & 31 & 27 \\
\hline 30 & 29 & 34 & 36 & 25 \\
\hline 30 & 32 & 38 & 35 & 40 \\
\hline 29 & 32 & 31 & 26 & 26 \\
\hline 32 & 26 & 30 & 40 & 32 \\
\hline 39 & 37 & 25 & 29 & 34 \\
\hline
\end{tabular}

The point estimate of the population mean is [tex]$\square$[/tex], and the point estimate of the proportion of defective units is [tex]$\square$[/tex].

Sagot :

GinnyAnswer
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].
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