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 26 days, it is considered defective. The table gives the sample data that a quality control manager collected.
| 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 |
The point estimate of the population mean is $\square$ and the point estimate of the proportion of defective units is $\square$
Answer (1)
Calculate the sample mean: 40 1292 = 32.3 .
Identify the number of defective units (lifespan < 26 days): 2.
Calculate the proportion of defective units: 40 2 = 0.05 .
The point estimate of the population mean is 32.3 and the point estimate of the proportion of defective units is 0.05 .
Explanation
Understand the problem and provided data We are given a sample of 40 units from a company's production, and we need to find the point estimate of the population mean lifespan and the proportion of defective units. A defective unit is defined as one with a lifespan less than 26 days.
Calculate the sample mean First, we need to calculate the sample mean. We sum all the lifespan values and divide by the number of units in the sample (40). The sum of the lifespan values is: 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 = 1292 Then, we divide the sum by 40 to get the sample mean: 40 1292 = 32.3
Identify the defective units Next, we need to find the proportion of defective units in the sample. We count the number of units with a lifespan less than 26 days. From the data, the defective units are: 25, 25. There are 2 such values less than 26. Therefore, there are 2 defective units in the sample.
Calculate the proportion of defective units Now, we calculate the sample proportion of defective units by dividing the number of defective units by the sample size: 40 2 = 0.05
State the final answer The point estimate of the population mean is 32.3, and the point estimate of the proportion of defective units is 0.05.
Examples
In manufacturing, estimating the mean lifespan and proportion of defective units helps companies maintain quality control. For example, if a company knows that the mean lifespan of its product is decreasing or the proportion of defective units is increasing, it can take corrective actions to improve its manufacturing process. This could involve improving the quality of raw materials, refining the manufacturing techniques, or enhancing quality control measures. By closely monitoring these estimates, companies can ensure customer satisfaction and reduce costs associated with defective products.
