The average daily high temperature for the month of September in Los Angeles is 27 degrees Celsius (81 degrees Fahrenheit) with a standard deviation of 2.78 degrees Celsius. How hot are the hottest 10.03% days during September in Los Angeles? Provide your answer to two decimal places and appropriate rounding.
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 (.1003) and its Z score. Insert data into equation above.
To find out how hot the hottest 10.03% days are during September in Los Angeles, we will use the concept of standard deviation.
Step 1: Convert the given percentage to a Z-score
The percentage of the hottest days, 10.03%, corresponds to the area under the normal distribution curve. To find the corresponding Z-score, we need to calculate the inverse of the cumulative distribution function.
Using a standard normal distribution table or a calculator, we find that the Z-score corresponding to 10.03% is approximately -1.28.
Step 2: Calculate the temperature using the Z-score
Using the Z-score formula:
Z = (X - μ) / σ
where:
Z is the Z-score,
X is the temperature we want to find,
μ is the mean temperature, and
σ is the standard deviation.
We rearrange the formula to solve for X:
X = Z * σ + μ
Plugging in the values from the question:
Z = -1.28
σ = 2.78 degrees Celsius
μ = 27 degrees Celsius
X = -1.28 * 2.78 + 27
Calculating this, we find:
X ≈ 23.55 degrees Celsius (rounded to two decimal places)
Therefore, the hottest 10.03% days during September in Los Angeles would be approximately 23.55 degrees Celsius.