Answer to Question #204037 in Statistics and Probability for maryam riaz

Question

(c) A group of amateur athletes ran 1000 m. The mode, median, and mean times were 230 seconds, 235 seconds, and 240 seconds respectively. The standard deviation of the times is 40 seconds.

i. Explain why the times cannot be normally distributed.

ii. Sketch a possible curve to show the distribution of the times.

Thirty of these athletes are selected at random. Find the probability that the average of their times is more than 255 seconds. Explain why it is possible to use the normal distribution to find this probability.

Answer 1

i. The mean, median, and mode of a normal distribution are equal.

Since mode =230 seconds, median = 235 seconds, and mean = 240 seconds, then the times cannot be normally distributed.

ii.

$$mode<median<mean$$

Right-skewed (positive skewness) distribution

iii.

The central limit theorem states that if you have a population with mean $\mu$ and standard deviation $\sigma$ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed.

This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large (usually $n\geq30$).

Let $X=$ the average of times. Since $n=30$ we can use the cnetral limit theorem.

Then $X\sim N(\mu,\sigma^2/n).$

Given $\mu=240\ s, \sigma=40\ s, n=30.$

$$P(X>255)=1-P(\leq 255)$$

$$=1-P(Z\leq \dfrac{255-240}{40/\sqrt{30}})\approx1-P(Z\leq2.05396)$$

$$\approx0.0200$$