- Z-test for the mean
- By hand (although we'll still use the TI-83 to get the Z critical value)
- With the TI-83
`Z-Test`

function

- Z-test for the proportion
- By hand, but using the TI-83 to get the Z critical value
- With the TI-83
`1-PropZTest`

function

*t*-test for the mean (with summary statistics given)- By hand (and using the TI-83 to get the
*t*critical value - we need to use the`EQUATION SOLVER`

for this) - With the TI-83
`T-Test`

function

- By hand (and using the TI-83 to get the
*t*-test for the mean (with "raw" data given)- By hand (using the TI-83 to get the
*t*critical value - and the summary statistics) - With the TI-83
`T-Test`

function

- By hand (using the TI-83 to get the

- 1.
- Consider the following exercise:
The policy of a particular bank branch is that its ATMs must be
stocked with enough cash to satisfy customers making withdrawals
over an entire weekend. At this branch the expected (i.e., population)
average amount of money withdrawn from ATM machines per customer
transaction over the weekend is $160 with an expected (i.e.,
population) standard deviation of $30. Suppose that a random sample
of 36 customer transactions is examined and it is observed that the
sample mean withdrawal is $172.
Note that the standard deviation that we are to use for this problem
did
*not*come from the sample. Therefore, this will be a**Z**-test, not a-test. For this problem, we have ,*t**n*= 36, and .- (a)
- State the null and alternative hypothesis.

This is not very clear, but apparently we are to check if the average*exceeds*$160, which would mean the ATMs are not "stocked with enough cash." This is what we will try to show, so it should go in the alternative hypothesis. Therefore the hypotheses would be:

- (b)
- At the .05 level of significance, using the critical value
approach to hypothesis testing, is there enough evidence to believe
that the true average withdrawal is greater than $160?

Let us get the test statistic:

`invNorm`

function with .95 as the argument, since, as your hand-drawn curve should clearly show, the area from to*Z*_{.05}is . The test statistic falls into the rejection region, i.e., 2.4 > 1.645, therefore we reject*H*_{0}. Yes, there is enough evidence that the average is more than $160. - (c)
- At the .05 level of significance, using the
*p*-value approach to hypothesis testing, is there enough evidence to believe that the true average withdrawal is greater than $160?

Now you should draw another Z-curve, this time shading the area to the right of the test statistic, 2.4. Alternatively, you could use the same curve you drew in (b) and shade the new area with a different color pen. This would make it clear that the area we wish to obtain will be less than . The shaded area is of course the*p*-value, and we can get it with the`normalcdf`

function. You have done this before (in Chapter 4) when you needed a probability.*p*-value is smaller than , thus we reject*H*_{0}. - (d)
- Interpret the meaning of the
*p*-value in this problem.

The probability of obtaining a sample whose mean is $172 or more when*H*_{0}is true is .0082. - (e)
- Compare your conclusions in (b) and (c).

The conclusions are the same, of course. A "large" test statistic, one that is larger than the critical value, is associated with a "small"*p*-value, one that is smaller than alpha.

And the decision rule is:

Reject*H*_{0}if the test statistic falls in the rejection region [part (b)], or

Reject*H*_{0}if the*p*-value is less than alpha [part(c)]. - (f)
- Now let us re-do part (a) using the
`Z-Test`

function. Under`STAT TESTS`

, choose`Z-Test`

. We do not have a list of data for this problem, instead we have summary statistics from the textbook. Choose`Stats`

. Enter the information for this problem as you see below.

With the cursor on`Calculate`

, press`ENTER`

.*p*-value, and does not give the critical value.

Another useful option is`Draw`

. Go back and choose`Draw`

instead of`Calculate`

. The graph should match what you drew by hand. Actually, for "extreme" test statistics, the shading will not show up on the graph. This one shows up just barely!

- 2.
- Consider the following exercise.

The marketing manager for an automobile manufacturer is interested in determining the proportion of new compact-car owners who would have purchased a passenger-side inflatable air bag if it had been available for an additional cost of $300. The manager believes from previous information that the proportion is .30. Suppose that a survey of 200 new compact-car owners is selected and 79 indicate that they would have purchased the air bags.

Since this is a hypothesis test for the proportion, it will be a**Z**-test. For this problem, we have*n*= 200, and*p*_{s}= 79/200 = .395.- (a)
- At the .10 level of significance, is there enough evidence that the
population proportion is different from .30?

The hypotheses here will be*H*_{0}:*p*= .30

`invNorm`

function with .95 as the argument. The test statistic falls into the rejection region, i.e., 2.93176 > 1.645, therefore we reject*H*_{0}. Yes, there is enough evidence that the population proportion is different from .30. - (b)
- Compute the
*p*-value and interpret its meaning.

Now you should draw another Z-curve, this time shading the area to the right (and left!) of the test statistic, . Of course, you could use the same curve you drew in (a) and shade the new area with a different color. This would make it easy to see that the area we wish to obtain will be less than . The shaded area is the*p*-value, and we get it with the`normalcdf`

function, like before.

. This*p*-value is smaller than , thus we reject*H*_{0}.

- (c)
- What is your answer to (a) if 70 new owners indicated that they
would have purchased the air bags?

Now we have*p*_{s}= 70/200 = .35. The test statistic changes to:

*H*_{0}. No, there is no enough evidence that the population proportion is different from .30.

Now, the*p*-value should be larger than . Indeed, . Do not reject*H*_{0}. - (d)
- Now let us re-do part (a) using the
`1-PropZTest`

function. Under`STAT TESTS`

, choose`1-PropZTest`

. Enter the information for this problem as you see below.

With the cursor on`Calculate`

, press`ENTER`

.*p*-value, and does not give the critical value.

Again, go back and choose`Draw`

. For this one, the shading will not show up on the graph.

Let us use the`1-PropZTest`

function for part (c).

- 3.
- Consider the following exercise:
The director of admissions at a large university advises parents
of incoming students about the cost of textbooks during a typical
semester. A sample of 100 students enrolled in the university
indicates a sample average cost of $315.40 with a sample
standard deviation of $43.20.
Note that the standard deviation
*did*come from the sample. Therefore, this will be a-test, not a*t***Z**-test. For this problem, we have*s*= 43.20,*n*= 100, and .- (a)
- Using the .10 level of significance, is there enough evidence that the
population average is above $300?

We are asked if the data shows that the mean is greater than $300. This will be the alternative hypothesis. Thus the hypotheses here will be

*t*-curve (just draw a symmetric bell-shaped curve like usual) and shade the last 10% of the right tail. We need the*t*critical value associated with this, which is . We get this value with the`EQUATION SOLVER`

, just like we did for*t*confidence intervals. Under the`MATH`

menu, choose`Solver`

. Recall the method explained on the confidence intervals example page for finding*t*critical values. This equation should still be in your calculator, but if not, enter in variables for the arguments like you see below.

`L`

is for*Lower bound*,`U`

is for*Upper bound*,`D`

is for*Degrees of freedom*, and`A`

is for*Area*.`L`

, so enter zero on the first line for a "guess." The upper bound is , so enter`1E99`

for`U`

. The degrees of freedom for this problem are n - 1 = 100 - 1 = 99, and we want the area between the lower and upper bound to be . With the cursor on the`L=0`

line, press`SOLVE`

(`ALPHA ENTER`

). Remember that this calculation takes about 15-20 seconds.*H*_{0}. Yes, there is enough evidence that the population average is above $300.

Now let us get the*p*-value.

Draw another*t*-curve, this time shading the area to the right of the test statistic, 3.5648. Again, you could use the same curve you drew in (a) and shade the new area with a different color. The shaded area is the*p*-value, and we get it with the`tcdf`

function.

This*p*-value is smaller than , thus we reject*H*_{0}.

- (b)
- What is your answer in (a) if the standard deviation is $75
and the .05 level of significance is used?

Now, s = 75 and . The test statistic will change to:

`EQUATION SOLVER`

and make the change to the**area**part of our equation.*H*_{0}.

Now let us get the*p*-value.

This*p*-value is smaller than , thus we reject*H*_{0}.

- (c)
- What is your answer in (a) if the sample average is $305.11?

Now, . The test statistic will change to:

*H*_{0}.

Now let us get the*p*-value.

Do not reject*H*_{0}.

- (d)
- Now let us re-do part (a) using the
`T-Test`

function. Under`STAT TESTS`

, choose`T-Test`

. We do not have a list of data for this problem, instead we have summary statistics from the textbook. Choose`Stats`

. Enter the information for this problem as you see below. With the cursor on`Calculate`

, press`ENTER`

.*p*-value, but not the critical value.

`T-Test`

also has the`Draw`

option. Go back and choose`Draw`

instead of`Calculate`

. Like before, our test statistic is too extreme to be shown on the screen. Imagine it off-screen to the right.

- 4.
- Consider the following exercise.

A manufacturer claims that the average capacity of a certain type of battery the company produces is at least 140 ampere-hours. An independent consumer protection agency wishes to test the credibility of the manufacturer's claim and measures the capacity of 20 batteries from a recently produced batch. The results, in ampere-hours, are as follows:137.4 140.0 138.8 139.1 144.4 139.2 141.8 137.3 133.5 138.2 141.1 139.7 136.7 136.3 135.6 138.0 140.9 140.6 136.7 134.1

We were not given a mean or standard deviation; we'll have to get them ourselves from the data. Of course, the standard deviation we get will be a sample standard deviation, which makes this a-test, not a*t***Z**-test. Enter the data into your calculator, into`L1`

, say. Obtain the summary statistics from`STAT CALC 1-Var Stats`

.*s*= 2.6589, and*n*= 20.- (a)
- Using the .05 level of significance, is there enough evidence that the
manufacturer's claim is being overstated?

The claim is that the average capacity is at least 140, and we will try to show that it is in fact less than 140. Thus the hypotheses here will be

*t*-curve and shade the leftmost 5% of the left tail. We need the*t*critical value associated with this, which is . We get this*t*critical value with the`EQUATION SOLVER`

, just like we did for the last problem.*H*_{0}. Yes, there is enough evidence that the manufacturer's claim is overstated.

Now let us get the*p*-value.

Draw another*t*-curve, this time shading the area to the left of the test statistic, -2.5734. Again, you could use the same curve you drew in (a) and shade the new area with a different color. The shaded area is the*p*-value, and we get it with the`tcdf`

function.

This*p*-value is smaller than , thus we reject*H*_{0}.

- (b)
- What assumption must hold in order to perform the test in (a)?

The population of battery capacities must be (approximately) normally distributed. - (c)
- Evaluate this assumption through a graphical approach.

Okay, let us see how likely it is that this data came from a normal population. First, let us look at a boxplot. This is very symmetric. Looks good so far.

Now let us look at a normal probability plot for this data. This is a very straight line. Both the boxplot and the normal probability plot seem to be telling us that our data does indeed come from a normal population. - (d)
- What is your answer in (a) if the last two values are 146.7 and 144.1
instead of 136.7 and 134.1?

Go to`STAT Edit`

and change the last two values.

*not*fall into the rejection region, i.e., . Therefore, we do not reject*H*_{0}.

For the*p*-value, shade the area under the*t*-curve to the left of the test statistic, -0.7466.

Do not reject*H*_{0}.

- (e)
- Now, instead of changing the data back to what it was originally
in part (a), let us leave it as it is and
re-do part (d) using the
`T-Test`

function.

This time, we*do*have a list of data, so choose`Data`

. Enter the information for this problem as you see below. With the cursor on`Calculate`

, press`ENTER`

.

Again, go back and choose`Draw`

. For this one, the shading is visible.