Question

The annual per capita consumption of bottled water was 34.5 gallons. Assume that the per capita consumption of bottled water is approximately normally distributed with a mean of 34.5 and a standard deviation of 11 gallons. a. What is the probability that someone consumed more than 35 gallons of bottled​ water? b. What is the probability that someone consumed between 25 and 35 gallons of bottled​ water? c. What is the probability that someone consumed less than 25 gallons of bottled​ water? d. 97.5​% of people consumed less than how many gallons of bottled​ water?

Answer 1

Answer:

a) 0.4819

b) 0.3242

c) 0.1939

d) 56.06 gallons            

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 34.5 gallons

Standard Deviation, σ = 11 gallons

We are given that the distribution of consumption of water is a bell shaped distribution that is a normal distribution.

Formula:

$z_{score} = \displaystyle\frac{x-\mu}{\sigma}$

a) P(consumed more than 35 gallons of bottled​ water)

P(x > 35)

$P( x > 35) = P( z > \displaystyle\frac{35 - 34.5}{11}) = P(z > 0.04545)$

$= 1 - P(z \leq 0.04545)$

Calculation the value from standard normal z table, we have,  

$P(x > 35) = 1 - 0.5181 = 0.4819 = 48.19\%$

b) P(consumed between 25 and 35 gallons of bottled​ water)

$P(25 \leq x \leq 35) \\\\= P(\displaystyle\frac{25 - 34.5}{11} \leq z \leq \displaystyle\frac{35-34.5}{11}) \\\\= P(-0.8636 \leq z \leq 0.0454)\\\\= P(z \leq 0.0454) - P(z < -0.8636)\\= 0.5181- 0.1939 =0.3242= 32.42\%$

$P(25 \leq x \leq 35) = 32.42\%$

c) P(consumed less than 25 gallons of bottled​ water)

$P( x < 25) = P( z < \displaystyle\frac{25 - 34.5}{11}) = P(z < -0.8636)$

Calculation the value from standard normal z table, we have,  

$P(x < 25) = 0.1939= 19.39\%$

d) We have to find the value of x such that the probability is 0.975

$P( X < x) = P( z < \displaystyle\frac{x - 34.5}{1})=0.975$  

Calculation the value from standard normal z table, we have,  

$\displaystyle\frac{x - 34.5}{11} = 1.960\\\\x = 56.06$

97.5​% of people consumed less than 56.06 gallons of bottled​ water.

Answer 2

Answer:

(a) 0.48006

(b) 0.3251

(c) 0.19489

(d) 97.5​% of people consumed less than 56 gallons of bottled​ water.

Step-by-step explanation:

We are given that the per capita consumption of bottled water is approximately normally distributed with a mean of 34.5 and a standard deviation of 11 gallons.

Let X = per capita consumption of bottled water

So, X ~ N($\mu = 34.5,\sigma^{2} =11^{2}$)

The z score probability distribution is given by;

            Z = $\frac{X-\mu}{\sigma}$ ~ N(0,1)

(a) Probability that someone consumed more than 35 gallons of bottled​ water = P(X > 35)

   P(X > 35) = P( $\frac{X-\mu}{\sigma}$ > $\frac{35-34.5}{11}$ ) = P(Z > 0.05) = 1 - P(Z $\leq$ 0.05)

                                                   = 1 - 0.51994 = 0.48006

(b) Probability that someone consumed between 25 and 35 gallons of bottled​ water = P(25 < X < 35)

   P(25 < X < 35) = P(X < 35) - P(X $\leq$ 25)

   P(X < 35) = P( $\frac{X-\mu}{\sigma}$ < $\frac{35-34.5}{11}$ ) = P(Z < 0.05) = 0.51994

   P(X $\leq$ 25) = P( $\frac{X-\mu}{\sigma}$ $\leq$ $\frac{25-34.5}{11}$ ) = P(Z $\leq$ -0.86) = 1 - P(Z < 0.86)

                                                   = 1 - 0.80511 = 0.19489

Therefore, P(25 < X < 35) = 0.51994 - 0.19489 = 0.3251

(c) Probability that someone consumed less than 25 gallons of bottled​ water = P(X < 25)

     P(X < 25) = P( $\frac{X-\mu}{\sigma}$ < $\frac{25-34.5}{11}$ ) = P(Z < -0.86) = 1 - P(Z $\leq$ 0.86)

                                                   = 1 - 0.80511 = 0.19489

(d) We have to find that 97.5​% of people consumed less than how many gallons of bottled​ water, which means ;

    P(X < x) = 0.975

    P( $\frac{X-\mu}{\sigma}$ > $\frac{x-34.5}{11}$ ) = 0.975

    P(Z > $\frac{x-34.5}{11}$ ) = 0.975

Now in the z table the critical value of X which have an area less than 0.975 is 1.96, i.e.;

           $\frac{x-34.5}{11}$ = 1.96

        $x$ - 34.5 = $1.96 \times 11$

                  $x$  = 34.5 + 21.56 = 56.06 ≈ 56 gallons of bottled water

So, 97.5​% of people consumed less than 56 gallons of bottled​ water.