Question

According to a recent article, 32% of drivers had driven drowsy in the past month. (Source: Thenationshealth.aphapublications.org/content/41/10/E52.fuII)

Suppose law enforcement officials are planning a survey of 1000 drivers to determine what proportion are driving drowsy.

a. Is whether a driver has driven drowsy in the past month qualitative or quantitative?

b. Would it be unusual if, in a random sample of 1000 drivers, 35% or more were driving drowsy in the last month? Include justifications for full credit.

c. What is the minimum number of drivers that must be sampled to be sure that the shape is approximately normal? Hint – use the formula

Answer 1

a)

We are provided numerical data, so whether a driver has driven drowsy in the past month is definitely quantitative.

b)

We are given:

We have to find the probability that

Using normal approximation to binomial as np>5, the sampling distribution of the sample proportion follows normal distribution.

i.e.

i.e.

Now

Since μ=0.32 and σ=0.01475 we have:

P ( X>0.35 )=P ( X−μ>0.35−0.32 )=P ( (X−μ)/σ>(0.35−0.32)/0.01475)

Since Z=(x−μ)/σ and (0.35−0.32)/0.01475=2.033898 we have:

P ( X>0.35 )=P ( Z>2.033898)

Use the standard normal table to conclude that:

P (Z>2.033898)= 1 - 0.979 = 0.020981

Since the probability is only 2.0981%, it is unusual.

c)

Minimum number of drivers can be calculated by using the following formula:

n = N*X / (X + N – 1)

where

X = Z_α/2² ­*p*(1-p) / MOE²

Here Z_α/2 is the critical value of the Normal distribution at α/2 (e.g. for a confidence level of 95%, α is 0.05 and the critical value is 1.96), MOE is the margin of error, p is the sample proportion, and N is the population size.

At 5% level of significance i.e. 95% Confidence level, Substituting values we get:

X = 1.96² * 0.32 * (1-0.32) / 0.95² = 334.372864

and

n = 1000 * 334.372864 / (334.372864+1000-1) = 250.77 = 251

Therefore the minimum sample size = 251 to be normal.

Please upvote, Thanks!