In: Math
According to the Kentucky Transportation Cabinet, an average of
167,000 vehicles per day crossed the Brent Spence Bridge into Ohio
in 2009. Give the state of disrepair the bridge is currently under,
a journalist would like to know if the mean traffic count has
increased over the past five years. Assume the population of all
traffic counts is bimodal with a standard deviation of 15,691
vehicles per day.
a. What conjecture would the journalist like to find support for in
this sample of vehicles?
b. A sample of 75 days is taken and the traffic counts are
recorded, completely describe the sampling distribution of the
sample mean number of vehicles crossing the Brent Spence Bridge.
Type out all supporting work.
c. The sample of 75 days had an average of 172,095.937 vehicles
crossing the bridge. What is the probability of observing a sample
mean of 172,095.937 vehicles or larger? Type out all supporting
work.
d. Based on the probability computed in part c, what can be
conclusion can be made about the conjecture? Explain.
e. If the number of sampled days was changed to 25, how would the
shape, mean, and standard deviation of the sampling distribution of
the sample mean traffic counts be affected?
a) The conjecture journalist would like to find support for is "mean traffic count is greater than 167,000 which was observed 5 years ago"
It can be defined by null hypothesis as:
vs alternative hypothesis:
b) A sample of 75 days is taken.
Thus sample size : n = 75
Here sample size is sufficiently large.
Hence sampling distribution of sample mean follows Normal Distribution with mean = and standard deviation =
Here, denote population mean and 's' denoted sample standard deviation. We use 's' because population standard deviation is unknown.
Here
Hence sampling distribution of sample means of traffic count follows Normal distribution with mean = 167000 and standard deviation of 1811.8406.
c) We have to find probability that, the sample of 75 days had an average of vehicles crossing the bridge at least 172,095.937.
i.e.
Convert into z score.
d) Probability of getting extreme mean count of traffic is very less. It is smaller than 5% significance level.Hence we reject null hypothesis. And support the claim that "mean traffic count is greater than 167,000 which was observed 5 years ago".
e) As sample size decreases it increases variability of shape of sampling distribution it becomes more platykurtic. Curve looks more flat and bulgy in the middle.
As mean still remains same standard deviation is increased from 1811.8406 to 3138.2
It follows t distribution assuming that parent population is Normally distributed and population variance is unknown.