Question

2. A local sports centre has showers with two temperature settings, warm and hot. On the warm setting, the water temperature may be modelled by a Normal distribution with mean 30°C and standard deviation 2°C (a) Using the model, find the probability that the next time the shower is used on the warm setting, the water temperature is (1) exactly 31°C (ii) more than 31°C The sports centre manager thinks that a water temperature of more than 33 °C is too high for the warm setting. She tests the water temperature on the warm setting on 5 randomly selected days. Given that the probability of the water temperature being more than 33 °C is 0.0668 (b) find the probability of the water temperature being more than 33 °C (1) on only the first of these 5 days, (ii) on more than 1 of these 5 days On the hot setting, the water temperature may be modelled by a Normal distribution with standard deviation 1.5°C The probability that the water temperature is more than 42°C is 0.0005 (c) Find the mean water temperature on this setting, giving your answer to 1 decimal place.

Answer 1

Que.2

a.

1.

$\\ P( X =31) = P\left ( \frac{ X - \mu}{\sigma} < \frac{31 - 30}{2} \right ) \\\\ \\ P(X = 31) = P (z=0.5) \\\\\\ P(X =31) = 0.3521$

2.

$\\ P( X > 31) = P\left ( \frac{ X - \mu}{\sigma} > \frac{31-30}{2} \right ) \\\\ \\ P( X > 31) = P (z>0.5) \\\\ \\ P( X > 31) = 1- P(z<0.5) \\\\ \\ P( X > 31) = 1-0.6915 \\\\\\ P(X >31) = 0.3085$

b.

Let X= number of days having water temperature more than 330C

X follows binomial distribution with parameter n=5 and p = 0.0668

The pmf of X is as follows:

i.

$P(X= x) = \binom{n}{x}p^xq^{(n-x)} \ \ \ \ \ x = 0,1,----,5$

$\\ P(X= 1) = \binom{5}{1}(0.0668)^1(1-0.0668)^{(5-1)}\\\\ P(X= 1) = 0.2533$

ii.

$\\ P(X> 1) = 1-P(X\leq 1)\\\\ P(X> 1) = 1-[P(0)+P(1)]\\\\ P(X> 1) = 1- [0.7077+0.2533] \\\\P(X> 1) = 0.039$

c.

For standard normal distribution,

P(Z > 3.29) = 0.0005

but,

$\\ z = \frac{X-\mu}{\sigma} \\ \\ \mu =X- \sigma z \\ \mu = 42 - 1.5 (3.29) \\ \mu =37.065$