Final answer:
To calculate the probability of the sample average wage falling between $30.00 and $31.00 for Swiss workers, you use the Central Limit Theorem to find the z-scores and refer to the standard normal distribution to determine the probability between those scores.
Step-by-step explanation:
The question involves finding the probability that the sample average of hourly wages for Swiss manufacturing workers will fall between two values. As given, the hourly wage in Switzerland is $30.67, with a standard deviation of $4.00. To find the required probability, we can apply the Central Limit Theorem (CLT) because the sample size of 39 is sufficiently large.
The first step is to calculate the standard error of the mean (SEM), which is the standard deviation divided by the square root of the sample size (n).
SEM = σ / √n = $4.00 / √39
SEM ≈ $0.64
Next, we find the z-scores for the given wage range. The z-score formula is (X - μ) / SEM, where X is the value of interest, μ is the mean, and SEM is the standard error of the mean.
Z-score for $30.00:
($30.00 - $30.67) / $0.64 ≈ -1.05
Z-score for $31.00:
($31.00 - $30.67) / $0.64 ≈ +0.52
Lastly, we look up these z-scores in the standard normal distribution table or use a normal distribution calculator to find the probabilities associated with them:
Probability of Z ≤ -1.05
Probability of Z ≤ +0.52
The probability that the sample average will be between $30.00 and $31.00 is the difference between these two probabilities.