Suppose the weekly salaries of elementary school teachers in GA are normally distributed with a mean of $871 and a standard deviation of $82. What is the probability (P) that the mean salary of 40 elementary school teachers is higher than $953 a week?

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BlackTom AI · Accepted Answer

We start by restating the problem and listing the options to ensure clarity before diving into the reasoning. Question: Suppose the weekly salaries of elementary school teachers in GA are normally distributed with a mean of $871 and a standard deviation of $82. What is the probability (P) that the mean salary of 40 elementary school teachers is higher than $953 a week? Answer options: ["P > 0.6", "0.3 < P < 0.6", "P < 0.1", "0.1 < P < 0.3"] Now, evaluate each option in turn: Option 1: P > 0.6 This would imply a very high probability that the sample mean exceeds 953. Given the population mean is 871 and the sampling distribution of the mean for n=40 has a standard error of 82/√40 ≈ 12.98, a threshold of 953 is about (953−871)/12.98 ≈ 6.32 standard deviations above the mean. In a normal distribution, being 6.3 SD above the mean corresponds to an extreme tail probability far less than 0.6. In fact, P(Z > 6.32) is extremely small, on the order of 10^-10 or smaller. Therefore, this option is not plausible. Option 2: 0.3 < P < 0.6 This would require a tail probability somewhere in the 0.3–0.6 range, which would place the threshold at around the 70th percentile or higher. Our calculated z-score of about 6.32 places us in the far upper tail with a probability far smaller than 0.3, making this option inconsistent with the actual distribution of the sample mean. Hence this option is incorrect. Option 3: P < 0.1 This aligns with the quantitative result. The standardized value z ≈ 6.32 puts the probability P(Z > 6.32) in the extreme tail, well below 0.1. While the exact tail probability is around 1e-10, the key takeaway is that the probability is exceedingly small, certainly less than 0.1. This option is consistent with the math. Option 4: 0.1 < P < 0.3 Similar to Option 2, this would place the probability in a moderate upper tail range. The actual P is far below 0.1 due to the very large z-score, so this option does not fit the calculated tail probability. In summary, the calculation shows that the sampling distribution of the mean has a tiny probability of exceeding 953, because the required mean is more than 6 standard errors away from the population mean. The resulting probability is an exceedingly small number, far less than 0.1, which makes Option 3 the one that matches the computed tail probability. The other options imply substantially larger tail probabilities that contradict the extreme z-score observed here.

Suppose the weekly salaries of elementary school teachers in GA are normally distributed with a mean of $871 and a standard deviation of $82. What is the probability (P) that the mean salary of 40 elementary school teachers is higher than $953 a week?单项选择题

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