Question

"Given that the mean daily high temperature for a certain season in a region is approximately normally distributed with a mean of 65 degrees Fahrenheit and a standard deviation of 4 degrees, answer the following:

a. Using appropriate technology (e.g., a calculator, statistical software), find the probability that one randomly selected day will have a high temperature between 58 and 73 degrees. Round your answer to the nearest thousandth.

b. Calculate the 40th percentile of the temperature distribution. Round your answer to the nearest tenth. Use the same technology for both parts of the question and specify which technology you used."

Answer 1

To find the probability of a randomly selected day having a temperature between 58 and 73 degrees, calculate the z-scores for these temperatures and use the standard normal distribution table. The probability is approximately 85.7%. To find the 40th percentile, find the corresponding z-score and then calculate the temperature value using the mean and standard deviation.

Step-by-step explanation:

To find the probability that one randomly selected day will have a high temperature between 58 and 73 degrees, we need to calculate the z-scores for these temperatures and then use the standard normal distribution table or a calculator or statistical software.

First, we calculate the z-scores for 58 and 73 using the formula: z = (x - mean) / standard deviation.

For 58, the z-score is: (58 - 65) / 4 = -1.75. For 73, the z-score is: (73 - 65) / 4 = 2.

Next, we use the standard normal distribution table or a calculator or statistical software to find the probabilities corresponding to these z-scores. Subtracting the cumulative probability for the lower z-score from the cumulative probability for the higher z-score, we find that the probability of a randomly selected day having a high temperature between 58 and 73 degrees is approximately 0.857 or 85.7%.

To calculate the 40th percentile of the temperature distribution, we need to find the z-score that corresponds to the 40th percentile. Using the standard normal distribution table or a calculator or statistical software, we find that the z-score corresponding to the 40th percentile is approximately -0.253. We then use the formula: x = z * standard deviation + mean to find the corresponding temperature value. Therefore, the 40th percentile of the temperature distribution is approximately 63.98 degrees Fahrenheit.