Question

The time needed for college students to complete a certain paper-and-pencil maze follows a Normal distribution with a mean of 35 seconds and a standard deviation of 2 seconds. (a) Answer the following questions: What is the random variable (X) of interest here? Describe the distribution of the random variable using statistical notation. Sketch the density curve for this distribution. [Remember to label the axes and indicate all important points.]

(b) You wish to see if the mean time is reduced by doing vigorous exercise before completing the maze, so you have a group of 16 college students exercise vigorously for 20 minutes and then complete the maze. It takes them an average of 34.5 seconds to complete the maze. Use this information to test the appropriate hypotheses with the "Four-Step Process". Conduct the test using a significance level of 10%. State any formulae you use before substituting values in them. Make sure to state any assumptions you make. Give a range for the P-value if the exact values are not available in the Statistical Tables. Group of answer choices

Answer 1

Given: The time needed for college students to complete a certain paper-and-pencil maze follows a Normal distribution with a mean of 35 seconds and a standard deviation of 2 seconds.

(a) Here, the random variable (X) of interest here is the ime needed for college students to complete a certain paper-and-pencil maze (in seconds).

As mentioned in the problem, X follows a Normal distribution with a mean of 35 seconds and a standard deviation of 2 seconds, i.e

X ~ N(35,2)

On graphing the density of this normally distributed random variable, we would obtain a symmetric bell-shaped curve peaked at X = 35 and the tails tapering towards the end. The graph drawn would be similar to:

(b) Given:

STEP 1: Stating the hypothesis:

We are asked to test:

Vs   

STEP 2: Setting up the criteria for decision:

As mentioned in the problem, we set the significance level at 10% level.If the p-value of the test falls below the fixed 0.10 level, we may reject the null hypothesis at 10% level, and fail to do so otherwise.

STEP 3: Computing the test statistic:

The appropriate statistical test to test the above (left-tailed) hypothesis would be a One sample Z test for mean, with test statistic given by:

with rejection region for the left tailed test, given by .

Substituting values in the test statistic:

= -1

The probability of obtaining a result ,at least as extreme as the one obtained, when the null hypothesis is true, called the p-value can be obtained from the normal table, by looking for the area corresponding to the test statistic (Z) obtained:

P-value =

Here, test statistic obtained is Z = -1:

From Normal table:

= 0.15866

STEP 4: Making a decision

Since, the p-value obtained for the test 0.15866 > 0.10 is not significant, we fail to reject the null hypothesis at 10% level. We may conclude that the data does not provide sufficient evidence to support the claim that the mean time is reduced by doing vigorous exercise before completing the maze.