Math  /  Data & Statistics

QuestionThe following data represent the high-temperature distribution for a summer month in a city for some of the last
Temperature 50-59 60-69 70-79 80-89 90-99 100-109
Days
4 322 1416 1515 483 11 (a) Approximate the mean and standard deviation for temperature. μ=80.3\mu=80.3 (Round to one decimal place as needed.) σ=8.3\sigma=8.3 (Round to one decimal place as needed.) (b) Use the frequency histogram of the data to verify that the distribution is bell shaped. Yes, the frequency histogram of the data is bell shaped. No, the frequency histogram of the data is not bell (c) According to the empirical rule, 95%95 \% of days in the month will be between what two temperatures? \square and \square (Round to one decimal place as needed. Use ascending order.)

Studdy Solution

STEP 1

1. The temperature data is grouped into intervals.
2. The mean (μ\mu) and standard deviation (σ\sigma) are given as 80.3 and 8.3, respectively.
3. The empirical rule states that 95% of data in a normal distribution lies within two standard deviations of the mean.

STEP 2

1. Calculate the approximate mean and standard deviation (already provided).
2. Verify if the distribution is bell-shaped using a frequency histogram.
3. Use the empirical rule to find the range of temperatures that cover 95% of the data.

STEP 3

The approximate mean (μ\mu) is given as:
μ=80.3 \mu = 80.3
The approximate standard deviation (σ\sigma) is given as:
σ=8.3 \sigma = 8.3

STEP 4

To verify if the distribution is bell-shaped, construct a frequency histogram using the given data:
- Temperature intervals: 50-59, 60-69, 70-79, 80-89, 90-99, 100-109 - Corresponding days: 4, 322, 1416, 1515, 483, 11
Examine the shape of the histogram. A bell-shaped distribution should have a single peak and be symmetric around the mean.

STEP 5

Apply the empirical rule to find the range of temperatures:
- Calculate the range for 95% of the data using the formula: μ±2σ\mu \pm 2\sigma
μ2σ=80.32(8.3)=80.316.6=63.7 \mu - 2\sigma = 80.3 - 2(8.3) = 80.3 - 16.6 = 63.7
μ+2σ=80.3+2(8.3)=80.3+16.6=96.9 \mu + 2\sigma = 80.3 + 2(8.3) = 80.3 + 16.6 = 96.9
Therefore, 95% of the days will be between:
63.7 and 96.9 \boxed{63.7} \text{ and } \boxed{96.9}

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