Final answer:
To calculate the probability that a simple random sample of 40 unemployed individuals will provide a sample mean within 1 week of the population mean, we can use the Central Limit Theorem and the z-score formula. The probability is 100%.
Step-by-step explanation:
To calculate the probability that a simple random sample of 40 unemployed individuals will provide a sample mean within 1 week of the population mean, we need to use the Central Limit Theorem. According to the Central Limit Theorem, the distribution of sample means from a population with any shape will approach a normal distribution as the sample size increases.
In this case, the population mean length of unemployment is 16.5 weeks, and the population standard deviation is 6 weeks. We want to find the probability that the sample mean is within 1 week of the population mean, so we can use the z-score formula to calculate this probability.
First, we need to calculate the standard error, which is the standard deviation of the sample mean. The standard error is equal to the population standard deviation divided by the square root of the sample size. So, the standard error is 6 / sqrt(40) = 0.9487.
Next, we calculate the z-score using the formula:
z = (x - μ) / σ
where x is the sample mean we want to find the probability of, μ is the population mean, and σ is the standard deviation (or standard error in this case).
z = (16.5 - 16.5) / 0.9487 = 0
Since the z-score is 0, we know that the sample mean is equal to the population mean, which means it is within 1 week of the population mean. Therefore, the probability is 1 or 100%.