Question

task: determine if your data is approximately normally distributed.
methods to use:

• visual inspection of histogram
• compare mean and median (should be close)
• empirical rule check:
• about 68% within 1 standard deviation of mean
• about 95% within 2 standard deviations of mean
• about 99.7% within 3 standard deviations of mean

what to write:

• calculate intervals for empirical rule
• count how many observations fall in each
• state whether data appears approximately normal
• provide reasoning for your conclusion

your name here elementary statistics december 3, 2023 9 / 25

Explanation:

Step1: Identify the discipline and subfield

The problem is about determining if data is normally distributed, which involves statistical methods. So the subfield is Statistics (under Mathematics).

Step2: Recall the Empirical Rule

The Empirical Rule states that for a normal distribution, about 68% of data lies within 1 standard deviation ($\mu \pm \sigma$), 95% within 2 standard deviations ($\mu \pm 2\sigma$), and 99.7% within 3 standard deviations ($\mu \pm 3\sigma$) of the mean ($\mu$).

Step3: Visual Inspection of Histogram

Check if the histogram of the data is roughly bell - shaped, symmetric around the center.

Step4: Compare Mean and Median

In a normal distribution, the mean ($\mu$) and median ($M$) are approximately equal. So calculate the mean and median of the data set and check if their values are close.

Step5: Calculate Intervals for Empirical Rule

First, find the mean ($\mu$) and standard deviation ($\sigma$) of the data set. Then calculate the intervals $\mu - \sigma$, $\mu+\sigma$; $\mu - 2\sigma$, $\mu + 2\sigma$; $\mu-3\sigma$, $\mu + 3\sigma$.

Step6: Count Observations in Each Interval

Count how many data points fall within each of the intervals calculated in Step 5. Then check if the percentages of data points in each interval are close to 68%, 95%, and 99.7% respectively.

Step7: State Conclusion

Based on the visual inspection of the histogram, the comparison of mean and median, and the Empirical Rule check (percentage of data in each interval), state whether the data appears to be approximately normally distributed. If the histogram is bell - shaped, mean and median are close, and the percentages of data in the Empirical Rule intervals are close to 68%, 95%, and 99.7%, then the data is approximately normally distributed; otherwise, it is not.

Answer:

The process to determine if data is approximately normally distributed involves: 1. Visually inspecting the histogram for a bell - shape. 2. Comparing mean and median (they should be close). 3. Calculating intervals using the Empirical Rule ($\mu\pm\sigma$, $\mu\pm 2\sigma$, $\mu\pm 3\sigma$), counting observations in each interval, and checking if the percentages match 68%, 95%, 99.7% respectively. Then state if the data is approximately normal with reasoning based on these checks.