Question

If 20% of the residents in a U.S. city prefer a white telephone over any other color available, what is the probability that among the next 1000 telephones installed in that city between 170 and 185 inclusive will be white?

Answer 1

The probability that among the next 1000 telephones installed in that city between 170 and 185 inclusive will be white is approximately 10.81%.

Step-by-step explanation:

We can use the normal approximation to the binomial distribution. First, we need to find the mean (μ) and standard deviation (σ) for the binomial distribution:

Mean (μ) = n * p

= 1000 * 0.20

= 200

Variance (σ2) = n * p * (1 - p)

= 1000 * 0.20 * 0.80

= 160

Standard Deviation (σ) = sqrt(Variance)

≈ sqrt(160)

= 12.65

Next, we convert the range 170 to 185 to a z-score:

zlower = (170 - 200) / 12.65

≈ -2.37

zupper = (185 - 200) / 12.65

≈ -1.19

We then look up these z-scores in a standard normal distribution table to find the probabilities:

• P(zlower) ≈ 0.0089
• P(zupper) ≈ 0.1170

Finally, to find the probability that the number of white telephones is between 170 and 185, we subtract the lower probability from the upper probability:

P(170 ≤ X ≤ 185) = P(zupper) - P(zlower)

= 0.1170 - 0.0089

≈ 0.1081 or 10.81%.