Looking at the means of the samples in the table, which of these values is least likely to be the mean of the population?
Answer (2)
Calculate the mean of the sample means: 16.8833.
Calculate the standard deviation of the sample means: 2.1598.
Calculate the z-scores for each value: 12.3 (-2.1221), 18.4 (0.7022), 19.0 (0.9800), 19.5 (1.2115).
The value with the largest absolute z-score is the least likely to be the population mean: 12.3 .
Explanation
Analyze the problem and data 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. To do this, we will calculate the mean and standard deviation of the sample means. Then, we will calculate the z-score for each of the given values. The value with the largest absolute z-score is the least likely to be the population mean.
Calculate the mean of the sample means First, calculate the mean of the sample means: ( 16.8 + 12.3 + 19.0 + 17.5 + 18.2 + 17.5 ) /6 = 16.8833
Calculate the standard deviation of the sample means Next, calculate the standard deviation of the sample means: σ = n ∑ i = 1 n ( x i − μ ) 2 = 6 ( 16.8 − 16.8833 ) 2 + ( 12.3 − 16.8833 ) 2 + ( 19.0 − 16.8833 ) 2 + ( 17.5 − 16.8833 ) 2 + ( 18.2 − 16.8833 ) 2 + ( 17.5 − 16.8833 ) 2 σ = 2.1598
Calculate the z-scores Now, calculate the z-score for each of the values 12.3, 18.4, 19.0, and 19.5: For 12.3: z = 2.1598 12.3 − 16.8833 = − 2.1221 For 18.4: z = 2.1598 18.4 − 16.8833 = 0.7022 For 19.0: z = 2.1598 19.0 − 16.8833 = 0.9800 For 19.5: z = 2.1598 19.5 − 16.8833 = 1.2115
Determine the largest absolute z-score and the least likely value Finally, determine the absolute z-scores:
Value
z-score
Absolute z-score
12.3
-2.1221
2.1221
18.4
0.7022
0.7022
19.0
0.9800
0.9800
19.5
1.2115
1.2115
The value with the largest absolute z-score is 12.3.
State the final answer Therefore, the value least likely to be the population mean is 12.3.
Examples
In quality control, sample means are often used to estimate the population mean of a production process. If we observe a sample mean that is very different from the expected population mean (i.e., has a large z-score), it could indicate a problem with the production process. For example, if we are producing bolts with a target diameter of 10mm, and we take several samples and find that one sample has a mean diameter significantly less than 10mm, it suggests that the machine producing the bolts may need adjustment.
The value least likely to be the population mean is 12.3, as it has the largest absolute z-score of 2.1221, indicating it is farthest from the mean of the sample means.
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