Question

83. Using the data from Lap 1 only, conduct a hypothesis test to determine if the mean time for completing a lap in races is the same as it is in practices.

Short answer

(a) The null hypothesis:$H_0:{\mu }_1={\mu }_2$

(b) The alternate hypothesis:$H_a:{\mu }_1\ne {\mu }_2$
(c) The difference between the mean times for completing a lap in races and practice is denoted by${\overline{X}}_1-{\overline{X}}_2$.
(d) Student's t distribution :$t_{20.32}$
(e) The test statistic: $-4.70$
(f) The $p$-value: $0.0001$
(g) Check student's solution:

(h)(i) Alpha: $0.05$
(ii) Decision: Reject the null hypothesis.
(iii) Reason for Decision: $p$-value $<$ alpha
(iv) Conclusion: There is sufficient evidence to conclude that the mean time for completing a lap in races is different from that in practices, at the $5\%$significance level,

Step-by-step solution

Given information

To conduct a hypothesis test to determine if the mean time for completing a lap in races is the same as it is in practices by using the data from Lap 1 only.

Explanation

(a) The null hypothesis is defined as follows:
$H_0:{\mu }_1={\mu }_2$
(b) The alternate hypothesis is defined as follows:
$H_a:{\mu }_1\ne {\mu }_2$
(c) The difference between the mean times for completing a lap in races and practice is denoted by $\overline{X_1}-\overline{X_2}$.
(d) Student's t distribution is $t_{20.32}$.
(e) For the mean difference, the paired test static as follows:

After ENTER all of the values, the OUTPUT will look like this:

Hence, the test statistic as: $-4.70$.

Explanation

(f) The $p$-value from the output is $0.0001$

(g) Using the data from the last exercise, create a mental picture of the situation.
The horizontal axis should be clearly labelled and scaled, and the $p$-value-corresponding region(s) on the graph should be shaded.

(h)

(i) $\alpha =0.05$

(ii) Decision: Null hypothesis is rejected.

(iii) $\alpha >p$-value

(iv) Conclusion: There is sufficient evidence to conclude that the mean time for completing a lap in races is different from that in practices at the $5\%$significance level.