Answer to Question #167646 in Statistics and Probability for Hafiz Faizal

Question

Reporting summary measures such as the mean, median, and standard deviation has become very common in modern life. Many companies, government agencies will report these descriptive measures of a variable, but they will rarely provide information on the shape of the distribution of that variable. In previous tutorials, you have learned some basic properties of some distributions that can help you to decide if a specific type of distribution is a good fit for a set of data. According to the National Diet and Nutrition Survey: Adults Aged 19 to 64, British men spend an average of 2.15 hours per day in moderate or high intensity physical activity. The standard deviation of these activity times for this sample was 3.59 hours. Can we infer that these activity times could follow a normal distribution? The following may provide an answer.

a. Sketch a normal curve marking the points representing 1, 2, and 3 standard deviations above and below the mean, and calculate the values at these points using a mean of 2.15 hours and standard deviation of 3.59 hours.

b. Examine the curve with your calculations. Explain why it is impossible for this distribution to be normal based on your graph and calculations.

c. Considering the variable being measured, is it more likely that the distribution is skewed to the left or that it is skewed to the right? Explain why.

d. Suppose that the standard deviation for this sample was 0.70 hours instead of 3.59 hours, which make it numerically possible for the distribution to be normal. Again, considering the variable being measured, explain why the normal distribution is still not a logical choice for this distribution. e. In your opinion, is there any difference between the term statistics and statistic?

Answer 1

$a.\ \mu=2.15,\ \sigma=3.59\\ \text{P.d.f. }p(x)=\frac{1}{\sqrt{2\pi}(3.59)}e^{-\frac{(x-2.15)^2}{2(3.59)^2}}\\ \text{The graph of }p(x):$

$\mu\pm\sigma,\ \mu\pm2\sigma,\ \mu\pm3\sigma\text{ and the values of }p(x) \text{ at these points are:}$

$b.\text{ It is impossible for our distribution to be normal.}\\ \text{Our random variable } X\text{(values of activity times) is non-negative.}\\ \text{The graph of } p(x)\text{ should be symmetric with respect to mean }\mu.\\ \text{Using the three-sigma rule we can say that }\approx 99.7\%\text{ of all the values}\\ \text{are in the interval } (\mu-3\sigma, \mu+3\sigma), \text {i. e. in the interval }(-8.62, 12.92).\\ c.\text{ It is more likely that the distribution is skewed to the right}.\\ \text{Because our random variable is non-negative, mean }\mu=2.15 \text{ is quite small }\\ \text{and }\sigma=3.59\text{ is quite big}.\\ d.\text{ The normal distribution is still not a logical choice for this distribution.}\\ \text{The distribution of the population is not normal.}\\ \text{We consider all British men aged 19 to 64. The activity time of a man}\\ \text{aged 64 is much lower than the activity time of a man aged 19. So the}\\ \text{distribution will be skewed to the right or to the left (mean > mode or}\\ \text{mean < mode respectively)}.\\ e.\text{ Statistics is a science that concerns the analysis of experimental data.}\\ \text{Statistic is any function computed from values in a sample}.$