Question

The amount of soda in a 16-ounce can is normally distributed with a mean of 16 ounces and a standard deviation of .5 ounce. What is the probability that a randomly selected can will have less than 15.5 ounces? Round your answers to four decimal places.

Answer 1

Answer:

Probability that a randomly selected can will have less than 15.5 ounces is 0.1587.

Step-by-step explanation:

We are given that the amount of soda in a 16-ounce can is normally distributed with a mean of 16 ounces and a standard deviation of 0.5 ounce.

Let X = amount of soda

So, X ~ N($\mu=16,\sigma^{2} =0.5^{2}$)

The z-score probability distribution for normal distribution is given by;

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

where, $\mu$ = mean amount = 16 ounces

            $\sigma$ = standard deviation = 0.5 ounce

The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X.

So, the probability that a randomly selected can will have less than 15.5 ounces is given by = P(X < 15.5 ounces)

  P(X < 15.5 ounces) = P( $\frac{ X -\mu}{\sigma}$ < $\frac{ 15.5-16}{0.5}$ ) = P(Z < -1) = 1 - P(Z $\leq$ 1)

                                                                 = 1 - 0.8413 = 0.1587

Now, in the z table the P(Z $\leq$ x) or P(Z < x) is given. So, the above probability is calculated by looking at the value of x = 1 in the z table which has an area of 0.8413.

Hence, the probability that a randomly selected can will have less than 15.5 ounces is 0.1587.