Question

State police believe that 70% of the drivers traveling on a major interstate highway exceed the speed limit. They plan to set up a radar trap and check the speeds of 80 cars. Using the 68-95-99.7 Rule, draw and label the distribution of the proportion of these cars the police will observe speeding.

Answer 1

Answer:

a) From the empirical rule we know that about 68% of the samples are between:

$\mu_p -\sigma = 0.7-0.0512=0.6488$

$\mu_p +\sigma = 0.7+0.0512=0.7512$

95% of the sample proportions would be between:

$\mu_p -2\sigma = 0.7-2*0.0512=0.5976$

$\mu_p +2\sigma = 0.7+2*0.0512=0.8024$

And 99.7 % would be between these limits:

$\mu_p -3\sigma = 0.7-3*0.0512=0.5464$

$\mu_p +3\sigma = 0.7+3*0.0512=0.8536$

And the figure attached explain the results obtained.

b) i) Independence condition of all the cars

ii) np>10 , 80*0.7=56>10

n(1-p)= 80(1-0.7)=24>10

We have all the conditions so then the normal model can be used.

Step-by-step explanation:

Part a

For this case we assume that the true parameter of interest on this case is p= proportion of drivers traveling on a major interstate highway exceeding the spped limit. For this case the mean and the deviation for the proportion is given by:

$\mu_p = 0.7$

$\sigma_p =\sqrt{\frac{p(1-p)}{n}}=\sqrt{\frac{0.7(1-0.7)}{80}}=0.0512$

From the empirical rule we know that about 68% of the samples are between:

$\mu_p -\sigma = 0.7-0.0512=0.6488$

$\mu_p +\sigma = 0.7+0.0512=0.7512$

95% of the sample proportions would be between:

$\mu_p -2\sigma = 0.7-2*0.0512=0.5976$

$\mu_p +2\sigma = 0.7+2*0.0512=0.8024$

And 99.7 % would be between these limits:

$\mu_p -3\sigma = 0.7-3*0.0512=0.5464$

$\mu_p +3\sigma = 0.7+3*0.0512=0.8536$

And the figure attached explain the results obtained.

b) Do you think the appropriate conditions necessary for your analysis are met? Explain.

We assume the following conditions:

i) Independence condition of all the cars

ii) np>10 , 80*0.7=56>10

n(1-p)= 80(1-0.7)=24>10

We have all the conditions so then the normal model can be used.