Question

Problem 3) A random digit dialing telephone survey of 880 drivers asked, “Recalling the intersections on your most recent drive to work or school , were any of traffic lights red when you entered the intersections, i.e. did you run any red lights?” Of the 880 respondents, 171 admitted that yes, at least one light had been red.

a) What is the response variable? Is it categorical or quantitative?

b) Show that the normal approximation for p̂ is valid by verifying the three conditions. Include the arithmetic. (Remember that when we “assumed” that we knew π, we used nπ ≥ 10. In contrast, this problem is reality where we do not know π: we are trying to estimate π with a confidence interval. So, we use ?̂to estimate π in checking the condition.)

c) Estimate the population proportion of drivers who ran a red light on the way to work or school with a 95 percent confidence interval. (Round the standard deviation to 2 nonzero decimals. For example, if you calculate ??̂ to be 0.01234, round to 0.012.)

d) Interpret the confidence interval with a statement in the context of the problem.

Answer 1

a) What is the response variable? Is it categorical or quantitative?

Here we are surveying about whether people ran a red light at intersections or not. So the replies would be either yes or no. Since yes or no are non -numerical

Response variable : people ran a red light at intersections or not on most recent drive to work or school

Categorical

b) Show that the normal approximation for p̂ is valid by verifying the three conditions. Include the arithmetic. (Remember that when we “assumed” that we knew π, we used nπ ≥ 10. In contrast, this problem is reality where we do not know π: we are trying to estimate π with a confidence interval. So, we use ?̂to estimate π in checking the condition.)

1. Random and independent sample - random digit dialing telephone is used so satisfied.
2. n > 30 that n should be large. - here 880 people sample so n > 30
3. np > 10 - np = (880 ) (171 / 880) >10 . So this is also satisfied.

Here = x / n = 171 / 800

= 0.1943

c) Estimate the population proportion of drivers who ran a red light on the way to work or school with a 95 percent confidence interval. (Round the standard deviation to 2 nonzero decimals. For example, if you calculate ??̂ to be 0.01234, round to 0.012.)

Confidence interval

Where = 0.1943

   =1 - 0.95 = 0.05

Therefore the critical value at

=

= 1.96 .............using normal percentage tables with p = 0.025

Substituting the values we have

d) Interpret the confidence interval with a statement in the context of the problem.

CI are always used for approximating a parameter here population proportion. Since it is an approximation and not actual ,we are certain about it but with not 100% confidence.

Interpretation: We are 95% confident that true proportion of people running a red light on most recent drive to work or school is within (0.168 , 0.22) .