Consider the means of the samples shown in the table. Which value is least likely to be the mean of the population from which the samples were taken?
| Sample | Sample Mean |
|---|---|
| 1 | 15.2 |
| 2 | 17.1 |
| 3 | 16.9 |
| 4 | 12.2 |
| 5 | 18.0 |
| 6 | 16.3 |
| 7 | 17.4 |
A. 12.2
B. 15.3
C. 16.4
D. 17.5
Answer (2)
Calculate the mean of the sample means: x ˉ ≈ 16.157 .
Calculate the standard deviation of the sample means: s ≈ 1.957 .
Calculate the z-scores for each value: 12.2 (-2.022), 15.3 (-0.438), 16.4 (0.124), 17.5 (0.686).
The value with the largest absolute z-score is the least likely: 12.2 .
Explanation
Analyze the problem We are given a set of sample means and asked to determine which of the given values is least likely to be the population mean. The least likely value will be the one that is furthest from the center of the distribution of sample means.
Calculate the mean of sample means First, calculate the mean of the sample means. This will give us an estimate of the population mean. The sample means are: 15.2, 17.1, 16.9, 12.2, 18.0, 16.3, 17.4. The mean of these values is calculated as follows:
x ˉ = 7 15.2 + 17.1 + 16.9 + 12.2 + 18.0 + 16.3 + 17.4 = 7 113.1 ≈ 16.157
Calculate the standard deviation of sample means Next, calculate the standard deviation of the sample means. This will give us a measure of the spread of the sample means. The standard deviation is calculated as follows:
s = n − 1 ∑ i = 1 n ( x i − x ˉ ) 2
s = 7 − 1 ( 15.2 − 16.157 ) 2 + ( 17.1 − 16.157 ) 2 + ( 16.9 − 16.157 ) 2 + ( 12.2 − 16.157 ) 2 + ( 18.0 − 16.157 ) 2 + ( 16.3 − 16.157 ) 2 + ( 17.4 − 16.157 ) 2
s = 6 ( − 0.957 ) 2 + ( 0.943 ) 2 + ( 0.743 ) 2 + ( − 3.957 ) 2 + ( 1.843 ) 2 + ( 0.143 ) 2 + ( 1.243 ) 2
s = 6 0.915849 + 0.889249 + 0.552049 + 15.657849 + 3.396649 + 0.020449 + 1.545049
s = 6 22.977142 ≈ 3.82952367 ≈ 1.957
Calculate the z-scores Now, calculate the z-scores for each of the given values (12.2, 15.3, 16.4, 17.5). The z-score is calculated as follows:
z = s x − x ˉ
For 12.2: z = 1.957 12.2 − 16.157 ≈ 1.957 − 3.957 ≈ − 2.022
For 15.3: z = 1.957 15.3 − 16.157 ≈ 1.957 − 0.857 ≈ − 0.438
For 16.4: z = 1.957 16.4 − 16.157 ≈ 1.957 0.243 ≈ 0.124
For 17.5: z = 1.957 17.5 − 16.157 ≈ 1.957 1.343 ≈ 0.686
The absolute values of the z-scores are approximately 2.022, 0.438, 0.124, and 0.686.
Determine the least likely value The value with the largest absolute z-score is the least likely to be the population mean. In this case, the largest absolute z-score is 2.022, which corresponds to the value 12.2. Therefore, 12.2 is the least likely to be the population mean.
Final Answer The least likely value to be the population mean is 12.2.
Examples
In quality control, understanding the distribution of sample means helps determine if a production process is drifting away from its target. By calculating z-scores, engineers can identify which observed values are statistically unlikely, indicating a potential problem in the manufacturing process. For example, if the target mean diameter of bolts is 16.157 mm, and a sample has a mean of 12.2 mm, the z-score helps assess whether this sample indicates a significant deviation requiring investigation. This ensures products meet the required specifications and maintains quality standards.
The value least likely to be the mean of the population is 12.2, as it has the largest absolute z-score of approximately 2.022 compared to the other options. This indicates that it is farthest from the mean of the sample means. Therefore, option A, 12.2, is the correct choice.
;
