State police believe that 40% 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 the first 80 cars which pass the trap. a) Using the 68-95-99.7 rule, draw and label the distribution of the proportion of these cars the police will observe speeding. Which graph shows the distribution of the proportion with the intervals for 68%, 95%, and 99.7%.b) Do you think the appropriate conditions necessary for your analysis are met? Are the conditions necessary to use a normal model met?1. yes all conditions are met2. no the randomization and 10% conditions are met.3. no the 10% condition is not met4. no the randomization condition is not met5. no the randomization and success/failure conditions are not met6. no the success/failure condition is not met7. no the 10% and success/failure conditions are not met8. no none of the conditions are met

Step-by-step explanation:

Hello!

The vairable of interest is

X: number of drivers, out of 80 drivers traveling the interstate, that exceded the speed limit.

n= 80

p=0.40

This variable is discrete and has a normal distribution. The conditions that should be met to apply the CLT and approximate the sampling distribution to normal are:

n≥30 ⇒ Check

n*p≥5 ⇒ 80*0.40= 32 Check

n*q≥5 ⇒ 80*0.6= 48 Check

All conditions are met then ^p≈N(p*;(pq)/n)

a) Using the 68-95-99.7 rule, draw and label the distribution of the proportion of these cars the police will observe speeding. Which graph shows the distribution of the proportion with the intervals for 68%, 95%, and 99.7%.

The mean of the distribution is p= 0.4

The standard deviation of the distribution is √[(pq)/n]= √[(0.4*0.6)/80]= 0.05

Empirical rule:

μ-σ≤0.66≤μ+σ ⇒ 0.4-0.05 ≤0.66≤ 0.4*0.05  ⇒ 0.35 ≤0.66≤ 0.45

μ-2σ≤0.95≤μ+2σ ⇒ 0.4-0.1≤0.95≤0.4+0.1 ⇒ 0.3≤0.95≤0.5

μ-3σ≤0.997≤μ+3σ ⇒ 0.4-0.15≤0.997≤0.4+0.15 ⇒ 0.25≤0.997≤0.55

See graphic in attachment.

b) Do you think the appropriate conditions necessary for your analysis are met? Are the conditions necessary to use a normal model met?

1. yes all conditions are met

2. no the randomization and 10% conditions are met.

3. no the 10% condition is not met

4. no the randomization condition is not met

5. no the randomization and success/failure conditions are not met

6. no the success/failure condition is not met

7. no the 10% and success/failure conditions are not met8. no none of the conditions are met

f(x) has value 150 at x= 0

and 0 at x= 3

so d(f(x))/dt = slope = -150/3 = -50