The average teacher's salary in North Dakota is $41,027. Assume a normal distribution with standard deviation of $5200.
A. what is the probability that a randomly selected teachers salary is less than 40,000?
B. What is the probability that the mean for a sample of 80 teachers' salaries is greater than 42,000
A. Use same process as previous post.
B. Now we are talking about a distribution of means rather than raw scores.
Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.
To calculate these probabilities, we need to use the concept of the standard normal distribution, also known as the Z-distribution. By converting the given values to Z-scores, we can find the corresponding probabilities from a standard normal distribution table or using statistical software.
A. To find the probability that a randomly selected teacher's salary is less than $40,000, we need to calculate the Z-score:
Z = (X - μ) / σ
where X is the value we want to find the probability for, μ is the mean, and σ is the standard deviation.
Z = (40,000 - 41,027) / 5200
Z ≈ -0.197
Using a standard normal distribution table or software, we can find the probability associated with this Z-score. The probability of a Z-score of -0.197 or less is approximately 0.4207.
Therefore, the probability that a randomly selected teacher's salary is less than $40,000 is approximately 0.4207 or 42.07%.
B. To calculate the probability that the mean for a sample of 80 teachers' salaries is greater than $42,000, we need to calculate the Z-score for the sample mean using the formula:
Z = (X - μ) / (σ / √n)
where X is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
In this case:
X = 42,000
μ = 41,027
σ = 5200
n = 80
Z = (42,000 - 41,027) / (5200 / √80)
Z ≈ 1.544
Using a standard normal distribution table or software, we can find the probability associated with this Z-score. The probability of a Z-score of 1.544 or more is approximately 0.9384.
Therefore, the probability that the mean for a sample of 80 teachers' salaries is greater than $42,000 is approximately 0.9384 or 93.84%.