Chapter 23, Problem 37E

(a)

To determine

To plot the data and explain do you think the conditions for inference are satisfied.

Answer to Problem 37E

All conditions are met.

Explanation of Solution

In this question, psychology experiments sometimes involve testing the ability of rats to navigate mazes. Thus, the plot of the data is as:

  

Thus, the conditions to be satisfied are as:

Random condition: It is satisfied assuming that the brands were randomly selected.

Independent condition: It is satisfied since the sample of eleven brands is less than $10\%$ of the population of all brands.

Normal condition with outlier: It is not satisfied because the plot shows no outlier.

Normal condition without outlier: It is satisfied because the plot is nearly linear.

Thus, without outlier all the conditions and assumptions for inferences are met.

(b)

To determine

To test the hypothesis that the mean completion time for this maze is $60$ seconds and explain what is your conclusion.

Answer to Problem 37E

There is sufficient evidence to reject the claim of a mean completion time of $60$ seconds.

Explanation of Solution

In this question, psychology experiments sometimes involve testing the ability of rats to navigate mazes. It is given that: $n=21$ .

The mean of the data is as:

  $\begin{array}{l}\overline{x}=\frac{38.4+46.2+\mathrm{....}+46.22+56.3}{21}\\ =52.2095\end{array}$

The standard deviation is as:

  $\begin{array}{l}s=\sqrt{\frac{{(38.4-52.2095)}^2+\mathrm{....}+{(56.3-52.2095)}^2}{21-1}}\\ =13.5646\end{array}$

Now, let us define the hypothesis as:

  $\begin{array}{l}{H}_0:\mu =60\\ H_1:\mu \ne 60\end{array}$

Thus, the value of statistics is:

  $\begin{array}{l}t=\frac{\overline{x}-{\mu }_0}{\frac{s}{\sqrt{n}}}\\ =\frac{52.2095-60}{\frac{13.5646}{\sqrt{21}}}\\ =-2.63\end{array}$

Thus, the P-value will be as:

  $0.01<P<0.02$

If the P-value is smaller than the significance level, then the null hypothesis is rejected. Thus,

  $P<0.05=5\%\Rightarrow \text{Reject }{H}_0$

Thus, there is sufficient evidence to reject the claim of a mean completion time of $60$ seconds.

(c)

To determine

To eliminate the outlier and test the hypothesis again and explain your conclusion.

Answer to Problem 37E

There is sufficient evidence to reject the claim of a mean completion time of $60$ seconds.

Explanation of Solution

In this question, psychology experiments sometimes involve testing the ability of rats to navigate mazes. It is given that: $n=20$ .

The mean of the data is as:

  $\begin{array}{l}\overline{x}=\frac{38.4+46.2+\mathrm{....}+46.22+56.3}{20}\\ =50.13\end{array}$

The standard deviation is as:

  $\begin{array}{l}s=\sqrt{\frac{{(38.4-52.2095)}^2+\mathrm{....}+{(56.3-52.2095)}^2}{20-1}}\\ =9.9041\end{array}$

Now, let us define the hypothesis as:

  $\begin{array}{l}{H}_0:\mu =60\\ H_1:\mu \ne 60\end{array}$

Thus, the value of statistics is:

  $\begin{array}{l}t=\frac{\overline{x}-{\mu }_0}{\frac{s}{\sqrt{n}}}\\ =\frac{50.13-60}{\frac{9.9041}{\sqrt{20}}}\\ =-4.46\end{array}$

Thus, the P-value will be as:

  $P<0.01$

If the P-value is smaller than the significance level, then the null hypothesis is rejected. Thus,

  $P<0.05=5\%\Rightarrow \text{Reject }{H}_0$

Thus, there is sufficient evidence to reject the claim of a mean completion time of $60$ seconds.

(d)

To determine

To explain do you think this maze meets the “one-minute average” requirement.

Answer to Problem 37E

Yes, this maze meets the “one-minute average” requirement.

Explanation of Solution

In this question, psychology experiments sometimes involve testing the ability of rats to navigate mazes. Thus, conclusion in both exercise part (b) and part (c) is that there is sufficient evidence to reject the claim of a mean completion time of $60$ seconds.Thus the maze does not mean the "one-minute average" requirement.