A study performed by Bridge Ratings found that in 2006, 58% of respondents said they use a cell phone while traveling in a vehicle (the survey participants were restricted to individuals who spent at least one hour per day commuting in their vehicle). For a sample of 250 adults who spend at least one hour per day commuting in a vehicle, approximate the probability that at least 150 use a cell phone while traveling in a vehicle. Use the normal approximation to the binomial distribution.
To approximate the probability that at least 150 out of 250 adults use a cell phone while traveling in a vehicle, we can use the normal approximation to the binomial distribution. This involves using the mean and standard deviation of the binomial distribution to approximate the distribution with a normal distribution.
First, we need to calculate the mean (μ) and standard deviation (σ) of the binomial distribution. The mean can be calculated using the formula:
μ = n * p
where n is the number of trials (250) and p is the probability of success (probability of using a cell phone while traveling), which is given as 58% or 0.58.
μ = 250 * 0.58 = 145
The standard deviation can be calculated using the formula:
σ = sqrt(n * p * q)
where q is the probability of failure (1 - p).
q = 1 - p = 1 - 0.58 = 0.42
σ = sqrt(250 * 0.58 * 0.42) ≈ 8.84
Next, we can use the normal approximation to estimate the probability. We want to find the probability that at least 150 out of 250 adults use a cell phone while traveling. This is equivalent to finding the probability of the range from 149.5 to 250.5 (since we can't have a fraction of a person).
To convert this range into a standard normal distribution, we need to calculate the z-scores for the lower and upper limits of the range. The z-score can be calculated using the formula:
z = (x - μ) / σ
For the lower limit (149.5), the z-score is:
z_lower = (149.5 - 145) / 8.84 ≈ 0.51
For the upper limit (250.5), the z-score is:
z_upper = (250.5 - 145) / 8.84 ≈ 11.39
Using a standard normal distribution table or a calculator, we can find the corresponding probabilities for these z-scores.
The probability of the range from 149.5 to 250.5 can be approximated by finding the cumulative probability from the z-score -0.51 to 11.39.
P(149.5 ≤ x ≤ 250.5) ≈ P(-0.51 ≤ Z ≤ 11.39)
The probability can be approximated by subtracting the cumulative probability of the lower z-score from the cumulative probability of the higher z-score.
P(149.5 ≤ x ≤ 250.5) ≈ P(Z ≤ 11.39) - P(Z ≤ -0.51)
Look up the cumulative probability values in a standard normal distribution table or use a calculator to find these probabilities.
Finally, subtract the cumulative probability of the lower z-score from the cumulative probability of the higher z-score to get the approximate probability:
P(≥ 150) ≈ P(149.5 ≤ x ≤ 250.5) ≈ P(Z ≤ 11.39) - P(Z ≤ -0.51)
Note: The probability obtained using the normal approximation may not be exact, but it provides a reasonable estimate when the number of trials (250) is large and the probabilities are not too extreme.