Question

Question 1

We wish to see if, on average, traffic is moving at the posted speed limit of 65 miles per hour along a certain stretch of Interstate 70. On each of four randomly selected days, a randomly selected car is timed and the speed of the car is recorded. The observed times were:

70                    65                    70                    75

Assuming that speeds are normally distributed with mean m, is there evidence that the mean speed is not equal to the posted speed limit?

a. Check the needed conditions for both the test statistic and confidence interval. (Do not do a stemplot.)

b. State Ho and Ha.

c. Calculate the test statistic (if applicable – state the degrees of freedom)

d. Find the p-value.

e. What is the conclusion for this problem? Do you reject Ho?

f. Calculate the 95% confidence interval.

g. Interpret the confidence interval

Answer 1

a)

In theory, the data should be drawn from a normal distribution or it is a large sample (needto check that n ≥ 30 ). In practice, using the t-distribution is sufficiently robust provided that there is little skewness and no outliers in the data. Look at a graph of the data.
• The data must be reasonably random.
• The sample must be less than 10% of the population.

b)

Below are the null and alternative Hypothesis,
Null Hypothesis: μ = 65
Alternative Hypothesis: μ ≠ 65

c)

Test statistic,
t = (xbar - mu)/(s/sqrt(n))
t = (70 - 65)/(4.0825/sqrt(4))
t = 2.449

d)

P-value Approach
P-value = 0.0918

e)

As P-value >= 0.05, fail to reject null hypothesis

f)

sample mean, xbar = 70
sample standard deviation, s = 4.0825
sample size, n = 4
degrees of freedom, df = n - 1 = 3

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, tc = t(α/2, df) = 3.18

ME = tc * s/sqrt(n)
ME = 3.18 * 4.0825/sqrt(4)
ME = 6.491

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (70 - 3.18 * 4.0825/sqrt(4) , 70 + 3.18 * 4.0825/sqrt(4))
CI = (63.5088 , 76.4912)

g)

Therefore, based on the data provided, the 95% confidence interval for the population mean is 63.5088 < μ < 76.4912 which indicates that we are 95% confident that the true population mean μ is contained by the interval (63.5088 , 76.4912)

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