Question

The standard deviation for a certain professor's commute time is believed to be 45 seconds. The professor takes a random sample of 25 days. These days have an average commute time of 13.4 minutes and a standard deviation of 53 seconds. Assume the commute times are normally distributed. At the .1 significance level, conduct a full and appropriate hypothesis test for the professor.

What are the appropriate null and alternative hypotheses? HH0:?2=45 H1:?2?45

Identify the other values given in the problem: n=25 x- =13.4 s=53 a=.10

Calculate the value of the test statistic. Round your response to at least 2 decimal places. x2=______
What is the corresponding P-value for the test statistic? ________Round your response to at least 4 decimal places.

Make a decision: Since ? <P, we the null hypothesis H0.

Help write a summary of the results of this hypothesis test: enough evidence in this sample to conclude the standard deviation in commute time is 45 seconds at the ?= significance level because P= ______ .

Answer 1

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis H₀: ? = 45

Alternative hypothesis H_A: ? $\neq$ 45

Formulate an analysis plan. For this analysis, the significance level is 0.05.

Analyze sample data. Using sample data, the degrees of freedom (DF), and the test statistic (X²).

DF = n - 1 = 25 -1

D.F = 24

$\chi ^{2}=\frac{(n-1)s^{2}}{\sigma ^{2}_{0}}$

$\chi ^{2}=\frac{(25-1)\times (53)^{2}}{(45)^2}$

$\chi ^{2}= 33.292$

We use the Chi-Square Distribution Calculator to find P(?² > 33.292) = 0.098

Interpret results. Since the P-value (0.098) is less than the significance level (0.10), we have to reject the null hypothesis.

From the above test we do not have sufficient evidence in the favor of the claim that the standard deviation in commute time is 45 seconds.