In: Math
Question 6
In a recent sample of 84 used car sales costs, the sample mean was $6,425 with a sample standard deviation of $3,156.
Assume the underlying distribution is approximately normal.
Part I) Which distribution should you use for for determining a confidence interval for estimating the population mean for used car sales costs?
a Normal Distribution
b t Distribution
c Uniform
d Chi Sq distribution
Part II) What is the number of degrees of freedom (df) for this problem.
Par III) Define the random variable X by selecting the appropriate letter below.
a An individual data item randomly selected from the population, some times referred to as the parent population.
b The average of n individual data item randomly selected from the parent population. In this question n is 84.
Part IV) Define the random variable X ¯ (Xbar) by selecting the appropriate letter below.
c An individual data item randomly selected from the population, sometimes referred to as the parent population.
d The average of n individual data item randomly selected from the parent population. In this question n is 84.
Part V) Construct a 95% confidence interval for the population mean time wasted.
Enter your answers rounded to 0 decimal places (Enter answer as an integer).
In the two answer locations provided enter the lower bound of the confidence interval first followed by the upper bound.
Part VI)
What is meant by the term “95% confident” when constructing a confidence interval for a mean?
a. If we took repeated samples, approximately 95% of the samples would produce the same confidence interval.
b If we took repeated samples,
approximately 95% of the confidence intervals calculated from those
samples would contain the sample mean.
c If we took repeated samples, the sample
mean would equal the population mean in approximately 95% of the
samples
d If we took repeated samples, approximately 95% of the confidence intervals calculated from those samples would contain the true value of the population mean.
n = sample size = 84
Sample mean and sample standard deviation s = 3156
Part I ) The population standard deviation is unknown here so t distribution is used to determine the confidence interval for the population mean.
Part II) Degrees of freedom = n -1 = 84 - 1 = 83
Part III) For random variable X option a is correct since X is an individual.
a. An individual data item randomly selected from the population, some times referred to as the parent population.
Part IV) option b is correct for Xbar since Xbar is the average.
b. The average of n individual data item randomly selected from the parent population. In this question, n is 84.
Part V) 95% confidence interval for the population mean:
The formula to find the confidence interval for mean is,
Where is the t critical value at a given confidence level.
c = confidence level = 0.95
alpha = 1 - 0.95 = 0.05
Using t distribution table the t critical value using degrees of freedom 83 and area in two tail as 0.05 is 1.989
Lower bound = 5740 and upper bound = 7110
Part VI) Confidence interval contains the true value of the population mean.
Option d is correct.
d. If we took repeated samples, approximately 95% of the confidence intervals calculated from those samples would contain the true value of the population mean.