The average daily high temperatures in Lawton are approximately normally distributed with a mean of 58 degrees and a standard deviation of 12 degrees. Find the probabilities below:
The probability that the high would not exceed 65 degrees on any one day.
The probability that the average high for 7 randomly chosen days would be over 65 degrees.
1. Z = (score-mean)/SD
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.
2. Z = (score-mean)/SEm
SEm = SD/√n
Use same table.
To find the probabilities in this problem, we can use the Z-score formula and the standard normal distribution table. The Z-score measures how many standard deviations an observation is from the mean in a normal distribution.
1. Probability that the high does not exceed 65 degrees on any one day:
To find this probability, we need to calculate the Z-score for 65 degrees using the formula: Z = (X - μ) / σ, where X is the value (65), μ is the mean (58), and σ is the standard deviation (12).
Z = (65 - 58) / 12
= 7 / 12
≈ 0.5833
Next, we need to find the corresponding value in the standard normal distribution table for a Z-score of 0.5833. This value represents the probability of a random observation falling below 65 degrees.
Using a standard normal distribution table or a calculator that provides this functionality, we find that the corresponding probability is approximately 0.7174.
Therefore, the probability that the high does not exceed 65 degrees on any one day is approximately 0.7174.
2. Probability that the average high for 7 randomly chosen days would be over 65 degrees:
To find this probability, we need to consider the distribution of sample means. The mean of the sample means is equal to the population mean, which is 58 degrees. The standard deviation of the sample means, also known as the standard error, is calculated by dividing the population standard deviation by the square root of the sample size.
For 7 days, the standard error is σ / √n = 12 / √7 ≈ 4.536
Next, we calculate the Z-score for a sample mean of 65 degrees using the formula: Z = (X - μ) / σ, where X is the value (65), μ is the mean (58), and σ is the standard error (4.536).
Z = (65 - 58) / 4.536
≈ 1.5469
Using a standard normal distribution table or a calculator, we find that the corresponding probability for a Z-score of 1.5469 is approximately 0.9387.
Therefore, the probability that the average high for 7 randomly chosen days would be over 65 degrees is approximately 0.9387.