Annual household incomes in a certain region have a distribution that is skewed right, with mean
30.25 thousand dollars and a standard deviation of 4.50 thousand dollars. If a random sample of 100
households is taken, then the probability that the sample mean ¯
x will be within one thousand dollars
of the true population mean is about
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To find the probability that the sample mean ¯x is within one thousand dollars of the true population mean, we can use the Central Limit Theorem (CLT).
The CLT states that for a large enough sample size (in this case, 100 households), the distribution of sample means will approach a normal distribution regardless of the shape of the population distribution.
We are given that the population distribution is skewed right, but since the sample size is large enough (n > 30), we can assume that the distribution of sample means will be approximately normal.
First, we need to calculate the standard error of the mean (SE), which is the standard deviation of the sample means. It is calculated by dividing the population standard deviation by the square root of the sample size:
SE = σ / √(n)
where σ is the population standard deviation, and n is the sample size.
In this case, the population standard deviation σ is given as 4.50 thousand dollars, and the sample size n is 100. Plugging these values into the formula, we get:
SE = 4.50 / √(100)
= 4.50 / 10
= 0.45 thousand dollars
Next, we need to find the z-score for the range within one thousand dollars of the true population mean. The z-score represents the number of standard errors the sample mean ¯x is away from the population mean. Since the range is within one thousand dollars, we have:
Range = 1000 thousand dollars
Z-score = Range / SE
= 1000 / 0.45
= 2222.22 (approximately)
Using a standard normal distribution table or calculator, we can find the probability associated with the z-score of 2222.22. This probability represents the likelihood that the sample mean ¯x will be within one thousand dollars of the true population mean.
Note that the probability of such a high z-score is extremely close to 1 (almost certain). However, it is practically impossible to have an exact z-score value of 2222.22, as it falls far beyond standard deviations from the mean.