In the Dominican Republic in August, the distribution of daily high temperature is approximately normal with mean 86 degrees Fahrenheit (°F). Approximately 95% of all daily high temperatures are between 83°F and 89°F. What is the standard deviation of the distribution?
approximately 95% means approximately 2 s.d. from the mean
the s.d. is approximately ... (89º - 83º) / 4
Well, if approximately 95% of all daily high temperatures are between 83°F and 89°F, that means we have the middle 95% of the data. This is also known as the "normal range".
Since a normal distribution is symmetric, we know that the remaining 5% of the data is split evenly on both sides. So, we can calculate each side separately.
On the left side, we have 2.5% of the data. Since the mean is 86°F, we need to find the value that is 2.5% below the mean. That would be 86°F - 3°F = 83°F.
On the right side, we also have 2.5% of the data. Since the mean is 86°F, we need to find the value that is 2.5% above the mean. That would be 86°F + 3°F = 89°F.
So, we have the range from 83°F to 89°F, which spans 6°F. Since the standard deviation measures the spread of the data, we can divide this range by 4 (since each side is 2.5%) to get the standard deviation.
Therefore, the standard deviation of the distribution is 6°F / 4 = 1.5°F.
Hope that helps, and remember to always pack your sunscreen in the Dominican Republic!
To determine the standard deviation of the distribution, we can use the 68-95-99.7 rule, also known as the empirical rule. According to this rule, approximately 95% of all daily high temperatures fall within two standard deviations above and below the mean.
In this case, the mean daily high temperature is given as 86°F, and it is stated that 95% of temperatures are between 83°F and 89°F. This interval represents two standard deviations from the mean.
So, we can set up the equation:
83°F = 86°F - 2σ (lower end of interval)
89°F = 86°F + 2σ (upper end of interval)
Next, we solve for the standard deviation (σ).
83°F = 86°F - 2σ
-3°F = -2σ
σ = (-3°F) / (-2)
σ = 1.5°F
Therefore, the standard deviation of the distribution is 1.5°F.
To find the standard deviation of the distribution, we need to use the information given about the percentage of daily high temperatures within a certain range.
The given information states that approximately 95% of all daily high temperatures are between 83°F and 89°F. In a normal distribution, 95% of the data falls within two standard deviations from the mean.
1. First, calculate the range of values that fall within two standard deviations from the mean:
89°F - 86°F = 3°F
86°F - 83°F = 3°F
2. Divide the range by 4 to get the standard deviation:
3°F / 4 = 0.75°F
So, the standard deviation of the distribution would be approximately 0.75°F.