The following data represents the high-temperature distribution for a summer month in a city for some of the last 130 years. Treat the data as a population.

Temperature Days
50-59 4
60-69 309
70-79 1428
80-89 1485
90-99 393
100-109 8

According to the empirical rule, 95% of days in the months will be between what temperature? (Round to one decimal place as needed)

60 89

To determine the temperature range within which 95% of the days in the month will fall, we can make use of the empirical rule, also known as the 68-95-99.7 rule. This rule states that in a normal distribution:

- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.

To apply this rule, we first need to calculate the mean and standard deviation of the high-temperature distribution. We can do this using the provided data.

To calculate the mean (average) temperature:

(sum of (midpoint of each range * frequency)) / (sum of frequencies)

midpoint of 50-59 range = (50 + 59) / 2 = 54.5
midpoint of 60-69 range = (60 + 69) / 2 = 64.5
midpoint of 70-79 range = (70 + 79) / 2 = 74.5
midpoint of 80-89 range = (80 + 89) / 2 = 84.5
midpoint of 90-99 range = (90 + 99) / 2 = 94.5
midpoint of 100-109 range = (100 + 109) / 2 = 104.5

(sum of (54.5 * 4) + (64.5 * 309) + (74.5 * 1428) + (84.5 * 1485) + (94.5 * 393) + (104.5 * 8)) / (4 + 309 + 1428 + 1485 + 393 + 8) = mean

After calculating, we find the mean temperature to be approximately 80.3 (rounded to one decimal place).

Next, we calculate the standard deviation of the high-temperature distribution. This involves finding the variance first, which is calculated as follows:

- For each range, calculate the difference between the midpoint and the mean.
- Square the difference and multiply it by the frequency.
- Sum up all the values.
- Divide the sum by the total frequency.

(((((54.5 - mean)^2) * 4) + (((64.5 - mean)^2) * 309) + (((74.5 - mean)^2) * 1428) + (((84.5 - mean)^2) * 1485) + (((94.5 - mean)^2) * 393) + (((104.5 - mean)^2) * 8)) / (4 + 309 + 1428 + 1485 + 393 + 8)) = variance

After calculating, we find the variance to be approximately 79.5 (rounded to one decimal place).

Finally, the standard deviation (SD) is the square root of the variance.

√variance = SD

After calculating, we find the standard deviation to be approximately 8.9 (rounded to one decimal place).

Now we can apply the empirical rule to calculate the temperature range within which 95% of the days in the month will fall. Since we are looking for two standard deviations from the mean, we multiply the standard deviation by 2 and add/subtract the result to/from the mean.

Temperature Range = mean ± (2 * SD)

80.3 ± (2 * 8.9)

Thus, we can conclude that 95% of the days in the month will be between 62.5 and 98.1 degrees Fahrenheit (rounded to one decimal place).

To determine the temperature range that corresponds to 95% of the days in the month, we can use the empirical rule (also known as the 68-95-99.7 rule) which states that for a normal distribution:

- Approximately 68% of the data falls within one standard deviation of the mean
- Approximately 95% of the data falls within two standard deviations of the mean
- Approximately 99.7% of the data falls within three standard deviations of the mean

In this case, we need to find the temperature range that corresponds to the middle 95% of the distribution.

First, let's calculate the mean and standard deviation for the temperature distribution:

Mean (μ) = (50-59) * 4 + (60-69) * 309 + (70-79) * 1428 + (80-89) * 1485 + (90-99) * 393 + (100-109) * 8
= 340 + 19041 + 100104 + 129135 + 88470 + 872
= 347962

Next, let's calculate the standard deviation (σ):

Standard deviation (σ) = sqrt[ ( (50-59-347962)^2 * 4 ) + ( (60-69-347962)^2 * 309 ) + ( (70-79-347962)^2 * 1428 ) + ( (80-89-347962)^2 * 1485 ) + ( (90-99-347962)^2 * 393 ) + ( (100-109-347962)^2 * 8 ) ]
= sqrt[ (897812^2 * 4) + (857941^2 * 309) + (777939^2 * 1428) + (632827^2 * 1485) + (541828^2 * 393) + (162148^2 * 8) ]
= sqrt[ 3224842384 + 2294918108816 + 848171231984 + 2997244943325 + 147617661712 + 20972466944 ]
= sqrt[ 5816213182525 ]
≈ 2,411,816

Now, we can calculate the range that corresponds to 2 standard deviations from the mean:

Lower temperature limit = Mean - (2 * Standard deviation)
= 347962 - (2 * 2411816)
= 347962 - 4823632
= -4475670

Upper temperature limit = Mean + (2 * Standard deviation)
= 347962 + (2 * 2411816)
= 347962 + 4823632
= 5171594

Therefore, according to the empirical rule, approximately 95% of days in the month will have temperatures between -4475670 and 5171594 degrees. However, negative temperatures do not make sense in this context, so we can assume the lower limit to be 0. Therefore, 95% of days in the month will have temperatures between 0 and 5171594 degrees.