2. The daily maximum temperature (T) on an island has approximately a normal distribution with a mean of 78 degree (F) and a standard deviation of 5 degree F on what proportion of days is the maximum temperature:
a. Greater than 70 degrees F
b. Greater than 85 degrees F
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To find the proportion of days on which the maximum temperature is greater than a certain value, we can use the standard normal distribution table or a statistical calculator.
a. To find the proportion of days when the maximum temperature is greater than 70 degrees F, we need to standardize the value using the mean and standard deviation. The formula for standardizing is (X - mean)/standard deviation.
Standardized value = (70 - 78)/5 = -8/5 = -1.6
Now, we can use the standard normal distribution table to find the proportion of values greater than -1.6. Looking up the value in the table, we find that the area to the left of -1.6 is approximately 0.0559. Therefore, the proportion of days with a maximum temperature greater than 70 degrees F is approximately 1 - 0.0559 = 0.9441, or 94.41%.
b. To find the proportion of days when the maximum temperature is greater than 85 degrees F, we again need to standardize the value.
Standardized value = (85 - 78)/5 = 7/5 = 1.4
Using the standard normal distribution table, we find that the area to the left of 1.4 is approximately 0.9192. Therefore, the proportion of days with a maximum temperature greater than 85 degrees F is approximately 1 - 0.9192 = 0.0808, or 8.08%.
So, on approximately 94.41% of the days, the maximum temperature is greater than 70 degrees F, and on approximately 8.08% of the days, the maximum temperature is greater than 85 degrees F.