AP Statistics Practice Exam 1: answers and scoring guide

Example: "For volunteer students at this school, is the mean reading speed after one month higher for students who use the app than for students who do not?"

A good question names the group, the variable (reading speed) and the comparison (app or no app).

Give the volunteers the numbers 1 to 40.

Use a random number generator to pick 20 different numbers from 1 to 40 (skip repeats).

Those 20 students use the app; the other 20 do not. Both groups read the same texts and are measured the same way.

Random assignment spreads other things that affect reading speed (how much each student reads at home, age) evenly between the groups.

Then a clear difference in reading speed can be said to be caused by the app.

The results apply only to students like these volunteers, not to all 600 students.

Volunteers may already like reading more, so they may gain more (or less) from the app than other students.

To say something about all 600 students, the 40 should be a random sample of the school.

The question

Question: find the mean of 6, 7, 8, 5, 9, 7, 6 and 8.

The mean is the fair share.

We add them all, then share the total equally.

1 Add them all

We add all the numbers, from left to right.

So the total is 56.

2 Count them

Now we count the numbers.

There are 8.

3 Share equally

We share the total equally: 56 divided by 8.

56 divided by 8 is 7.

4 Check

Let's check: if every number were 7, 8 of them would make 7 times 8.

That is 56, the total.

So it is right!

Let's look back.

First: add them all.

Then: count them.

Then: share equally.

Last: check.

The question

Question: find the sample standard deviation of 6, 7, 8, 5, 9, 7, 6 and 8.

The standard deviation, SD, tells how far the numbers usually are from the mean.

1 Find the mean

First the mean: add them all, then divide by how many.

The total is 56.

56 divided by 8 is 7.

2 Distances from the mean

A table: how far each number is from the mean.

6 minus 7 is −1.

Negative: 6 is below the mean.

7 minus 7 is 0.

8 minus 7 is 1.

Above the mean.

Look: the distances add up to 0.

The ones below the mean cancel the ones above.

So adding the distances tells us nothing: it is always 0.

We need every distance to count as positive.

So we square each distance.

A square is never negative.

3 Square them

A new column: each distance squared.

−1 squared is −1 times −1.

Negative times negative is positive: 1.

And the rest.

4 Add the squares

We add the squares: 1 plus 0 plus 1 plus 4 plus 4 plus 0 plus 1 plus 1 is 12.

5 Divide

These numbers are a sample: only part of the whole group.

A sample is usually a bit closer to its own mean than the whole group is.

So we divide by one less than how many: n minus 1, where n is how many numbers.

That makes the answer a little bigger, to make up for it.

There are 8 numbers, so we divide by 8 minus 1, which is 7.

12 divided by 7 is 1.7143 and more digits.

Careful!

For a sample we divide by 7, not by 8.

6 Square root

We squared the distances, so the variance is in squared units.

The square root brings it back to the units of the numbers.

The square root of 1.7143… is 1.3093 and more digits.

We round to one more decimal place than the data, so the answer does not look more exact than the numbers.

So the SD is about 1.3.

So the numbers are usually about 1.3 away from the mean, 7.

7 Check

Let's check: 1.3 squared is 1.69.

That is about the variance, 1.71.

So it is right!

Let's look back.

First: find the mean.

Then: distances from the mean.

Then: square them.

Then: add the squares.

Then: divide.

Then: square root.

Last: check.

In order: 5, 6, 6, 7, 7, 8, 8, 9. Median = 7; Q1 = 6 (the median of 5, 6, 6, 7); Q3 = 8 (the median of 7, 8, 8, 9).

The data are about symmetric: the mean and the median are both 7.

z = (10 − 7)/1.3 ≈ 2.3.

10 hours is about 2.3 standard deviations above the mean sleep of the sample.

That is unusually long sleep for these students.

H0: p = 0.3 and Ha: p > 0.3.

p = the proportion of all working residents of the town who bike to work.

Random: the sample is random. 10%: 200 is less than 10% of the town's working residents.

Large counts: 200 × 0.3 = 60 and 200 × 0.7 = 140, both at least 10.

The question

Question: in a sample of 200, 75 say yes: at the 5% level, is the true proportion more than 0.3?

1 Write the hypotheses

H₀ is the "nothing new" idea: it has the equal sign.

Here: the proportion is 0.3.

Hₐ is what we want to show.

Here: the proportion is more than 0.3.

More than means only the right end counts.

One tail.

2 Big enough for the bell

The bell works if both counts are at least 5.

200 times 0.3 is 60.

200 times 0.7 is 140.

Both are big enough.

3 The test number

p prime, the sample's part: 75 divided by 200 is 0.375.

The SD uses p zero from H zero: 0.3 times 0.7, over 200, then the square root.

About 0.0324.

Careful!

We act as if H zero is true.

So the SD uses 0.3, not 0.375.

z is p prime minus p zero, over the SD.

0.375 minus 0.3 is 0.075.

Divided by 0.0324: about 2.31.

4 The p-value

The p-value is the chance of getting a sample at least this far out, if H zero is true.

On the bell of H zero: 0.3 in the middle, our sample at z = 2.31.

We shade the right end, from our z up.

The z-table gives the area to the left of z.

Row 2.3, column .01.

For z = 2.31: the row 2.3 and the column .01 meet at 0.9896.

The right end is 1 minus 0.9896: 0.0104.

5 Decide

The p-value, 0.0104, is less than alpha, 0.05.

Our sample would be rare if H zero were true.

So we reject H zero.

6 Say it in words

In words: at the 5% level, there is enough evidence that the proportion is more than 0.3.

7 Check

Let's check the other way, with the cut-off.

0.05 in the right end gives the cut-off 1.645, from the table, as in the cut-offs lesson.

z is 2.31: past the cut-off.

The same decision.

So it is right!

Let's look back.

First: write the hypotheses.

Then: big enough for the bell.

Then: the test number.

Then: the p-value.

Then: decide.

Then: say it in words.

Last: check.

The p-value ≈ 0.0104 is less than α = 0.05, so we reject H0.

There is convincing evidence that more than 30% of the town's working residents bike to work.

The question

Question: There are 8 games.

Each one is a success with probability 30%.

What is the probability of exactly 2 successes?

1 Binomial: yes or no

8 games.

Each one has the same chance, and one does not change the next.

Each is a success or not.

That is a binomial experiment.

n is the number of games: 8.

k is how many successes we want: 2.

p is the chance of a success each time: 0.3.

q is the chance of not: 1 minus p, 0.7.

2 Write the rule

The rule has two parts.

n C k counts the orders.

p to the power k, times q to the power n minus k, is the chance of one order.

We build each part, then multiply.

We put the numbers in.

3 Count the orders

8 C 2 counts the orders: which 2 of the 8 games are the successes.

8 times 7, over 2 times 1, is 28.

4 One order's chance

One order, like this one.

For one order we multiply the chances, one for each try, as in the and lessons: 0.3 for each success, and 0.7 for each failure.

0.09 times 0.117649 is 0.01058841.

5 Multiply

Every order has this same chance.

There are 28 orders, so we multiply by 28.

Careful!

That is only one order.

Successes can come in 28 different orders.

28 times 0.01058841 is 0.29647548.

To 4 decimal places: about 0.2965.

6 Check

Let's check: 28 orders, each with the same chance.

So the answer is right!

Let's look back.

First: binomial: yes or no.

Then: write the rule.

Then: count the orders.

Then: one order's chance.

Then: multiply.

Last: check.

Mean = np = 8 × 0.3 = 2.4 items.

SD = √(np(1 − p)) = √(8 × 0.3 × 0.7) = √1.68 ≈ 1.296 items.

The question

Question: X is normal, with mean 70 and SD 6: what is the probability that X is more than 80?

μ is the mean.

σ is the standard deviation, the SD.

1 Find z

z counts how many SDs x is from the mean: x minus the mean, over the SD.

Every normal bell has the same shape when we count in SDs.

So one table, the z-table, works for every bell.

80 minus 70 is 10.

Divided by 6, that is about 1.67.

2 Draw and shade

The bell: the mean, 70, in the middle.

80 is right of it.

We want X more than 80.

We shade that part.

The whole bell has area 1, and the shaded area is the probability.

3 Read the table

The z-table gives the area to the left of z.

Row 1.6, column .07.

For z = 1.67: the row 1.6 and the column .07 meet at 0.9525.

4 The area we want

Careful!

0.9525 is the area to the left.

We want the right side.

The whole bell is 1.

The right side is 1 minus the left: 1 minus 0.9525 is 0.0475.

5 Check

Let's check another way.

The bell is the same on both sides.

So the area right of 1.67 is the area left of −1.67.

The table at −1.67 gives 0.0475.

The same!

So it is right!

Let's look back.

First: find z.

Then: draw and shade.

Then: read the table.

Then: the area we want.

Last: check.

P(both) = 0.0475 × 0.0475 ≈ 0.0023.

This assumes the two weights are independent of each other.