Question

a hundred squash balls are tested by dropping from a height of 100 inches . a ball is fastt if it rises above 32 inches the average height of bounce was 30 inches and Standard deviation was 3/4.what is the chance of getting a fast standard ball

Answer 1

Conditions

a hundred squash balls are tested by dropping from a height of 100 inches. A ball is fast if it rises above 32 inches the average height of bounce was 30 inches and Standard deviation was 3/4. What is the chance of getting a fast standard ball

Solution

For solving this, we must assume, that the test of these balls has approximately normal distribution.

Then, we must find the following probability:

$$P \left(\mu > \bar {X} + z _ {\frac {1 + \alpha}{2}} \frac {\sigma}{\sqrt {n}}\right) = 1 - P (\mu \leq \bar {X} + z _ {\frac {1 + \alpha}{2}} \frac {\sigma}{\sqrt {n}})$$

We know, that

$$P \left(\bar {X} - z _ {\frac {1 + \alpha}{2}} \frac {\sigma}{\sqrt {n}} \leq \mu \leq \bar {X} + z _ {\frac {1 + \alpha}{2}} \frac {\sigma}{\sqrt {n}}\right) = \alpha$$

And

$$P \left(\bar {X} \leq \mu \leq \bar {X} + z _ {\frac {1 + \alpha}{2}} \frac {\sigma}{\sqrt {n}}\right) = \frac {\alpha}{2}$$ $$\bar {X} = 3 0$$ $$\sigma = \frac {3}{4}$$ $$n = 1 0 0$$ $$\mu \leq 3 0 + z _ {\frac {1 + \alpha}{2}} \frac {\frac {3}{4}}{1 0} = 3 0 + 0. 0 7 5 z _ {\frac {1 + \alpha}{2}} = 3 2$$ $$z _ {\frac {1 + \alpha}{2}} = \frac {2}{0 . 0 7 5}$$ $$z _ {\frac {1 + \alpha}{2}} \approx 2 6. 6$$

Now we have to look at the Laplace function values for For this quantile the probability of

$$P \left(\bar {X} - z _ {\frac {1 + \alpha}{2}} \frac {\sigma}{\sqrt {n}} \leq \mu \leq \bar {X} + z _ {\frac {1 + \alpha}{2}} \frac {\sigma}{\sqrt {n}}\right) \approx 1$$

Hence, the probability of

$$P \left(\mu > \bar {X} + z _ {\frac {1 + \alpha}{2}} \frac {\sigma}{\sqrt {n}}\right) \approx 0$$

The event of getting the fast ball is almost impossible.

**Answer: the probability is approximately 0**