In a certain city, the average monthly temperature is 30 degrees Celsius with a standard deviation of 3 degrees. What is the probability that the average temperature for a randomly selected month falls between 27 and 33 degrees?
Answer
To find the probability that a randomly selected score falls between these two z-scores, subtract the smaller probability from the larger one: \(0.8413 - 0.1587 = 0.6826\)
Step 1 ：First, we need to convert the temperature values to z-scores. The z-score is calculated using the formula: \(z = \frac{X - \mu}{\sigma}\), where X is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
Step 2 ：For X = 27, the z-score is: \(z = \frac{27 - 30}{3} = -1\)
Step 3 ：For X = 33, the z-score is: \(z = \frac{33 - 30}{3} = 1\)
Step 4 ：Next, we find the probabilities associated with these z-scores from the standard normal distribution table. The probability for z = -1 is 0.1587 and for z = 1 is 0.8413.
Step 5 ：To find the probability that a randomly selected score falls between these two z-scores, subtract the smaller probability from the larger one: \(0.8413 - 0.1587 = 0.6826\)
