Chpt 8 - Estimation of the Mean and Proportion

Question 1

What are inferential statistics?

Answer 1

Statistical methods that draw and measure the reliability of conclusions about a population based on information obtained from a sample of the population

Question 2

Describe the relationship between point estimation, a point estimator and a point

Answer 2

Point estimation is the statistical process of finding an approximate value of some parameter.

The point estimator of the population mean (parameter) is the sample mean (X̄).

The point estimate is the product of the process. So the point estimate of the population mean is a value of the mean of a sample (x̄).

Question 3

What is a confidence interval?

Answer 3

It is an estimated range of values (obtained based on a sample) which includes the population mean (μ) with certain probability

Question 4

What is the half length of a confidence interval for the population mean (μ) denoted as?

Answer 4

c

Question 5

What are the upper and lower limits of the confidence interval?

Answer 5

lower bound: x̄-c

higher bound: x̄+c

Question 6

What is a confidence level?

Answer 6

Quantifies the level of confidence that the population mean lies in the interval

Basically, its the percentage of the confidence intervals that contain the population mean

Question 7

Can we tell if the population mean is in the confidence interval, even if the confidence level is really high, say 99%?

Answer 7

No, it’s impossible

Question 8

What is the equation for a confidence level?

Answer 8

x̄ +/- (Z α/2) x (σ/√n)

Question 9

For the equation for the confidence interval, what is x̄?

Answer 9

The sample mean of a sample

Question 10

For the equation for the confidence interval, what is
Zα/2?

Answer 10

The z-score whose right area under the standard normal density curve is α/2

Question 11

For the equation for the confidence interval, what is σ?

Answer 11

The population standard deviation

Question 12

The volumes of children’s Tylenol are expected to have a mean 100mL, but the exact volume varies from bottle to bottle.

A random sample of 50 bottles are picked and the exact volume of each bottle is measured.

Suppose that the average volumes is 98 mL and the population standard deviation is 2mL.

Identify the sample size, sample mean, and the distribution of X̄?

Answer 12

n = 50

x̄ = 98

Because n>30, the distribution is approximately normal

Question 13

The volumes of children’s Tylenol are expected to have a mean 100mL, but the exact volume varies from bottle to bottle.

A random sample of 50 bottles are picked and the exact volume of each bottle is measured.

Suppose that the average volumes is 98 mL and the population standard deviation is 2mL.

How would we determine the lower and upper bounds of a 90% confidence interval?

Answer 13

Determine important values:
n = 50
x̄ = 98
σ = 2

Determine the Zα/2 value

(1 - α) 100% = 90%
1 - α = 0.9
α = 0.1
α/2 = 0.05
This is the right area, so we find the left area, 1-0.05 = 0.95
Then we go to table 2, find 0.95 in the middle values to find the z-score along the sides (1.645)

Determine σ/√n value
σ/√n = 2/√50 = 0.28284

Determine (Z α/2) x (σ/√n)
1.645 x 0.28284 = 0.4653

Determine upper limit:
x̄ + (Z α/2) x (σ/√n)
98 + 0.4653 = 98.4653

Determine lower limit:
x̄ - (Z α/2) x (σ/√n)
98 + 0.4653 = 97.5347

Question 14

A 95% confidence interval is (97.45, 98.55).

What does this mean?

Answer 14

We are 95% confident that the interval (97.45, 98.55) contains the population mean

Question 15

What is the value of c in a confidence interval?

Answer 15

(Z α/2) x (σ/√n)

Question 16

A wider confidence interval has a ______ confidence level than a narrower confidence level. However, it doesn’t mean that the confidence level is the higher the better because a narrower confidence interval makes the estimate more _____ than a wider confidence interval.

Answer 16

larger

precise

Question 17

How do we improve the estimation precision of a (1-α)100% confidence interval?

What can we do to accomplish this?

Answer 17

We need to tighten the confidence interval

We can decrease the margin of error, therefore decreasing the size of the confidence interval by increasing the sample size

Remember that the margin of error is from the mean to the end of the confidence interval so the margin x 2 is the confidence interval.

Question 18

What is the margin of error of a confidence level in relation to a confidence interval and what is the equation we use for a (1-α)100% confidence interval?

Answer 18

It’s half the length of the confidence interval

Zα/2 * σ/√n

So remember that for a confidence interval, we do this +/- to the sample mean to get the range of the interval so the margin of error is half this value which is the value from one end to the mean

Question 19

What is the equation to determine how large a sample size should be if we know σ?

Answer 19

n = ((Zα/2*σ)/Eo) squared

Remember:

Eo is the margin of error

Question 20

Children’s Tylenol ha a population standard deviation of 2mL and when n=50, the confidence level of 95%, the margin of error is 0.554. If we want to reduce the margin of error to 0.5 but keep the confidence level (still 95%), how large should sample size n be?

Answer 20

(1-α)100% = 95%
α = 0.05
α/2 = 0.025

Z0.025 = 0.975 = Zscore 1.96

n = ((Zα/2σ)/Eo) squared
= ((1.962)/0.5) squared
= 61.4

Round up (because we can’t measure half a bottle), so we need a sample size of 62

Question 21

When we are determining a confidence interval, what can be used if σ is unknown?

Answer 21

We can replace it with the sample standard deviation s

Question 22

What is the equation for the sample standard deviation?

Answer 22

s = square root of:

     (n-1)

Question 23

When constructing a confidence interval for μ and σ is unknown, we use an equation based on
x̄-μ
——
s/√n
What type of distribution is this?

Answer 23

It is a t-distribution with degrees of freedom (n-1)

Question 24

Is a t-distribution approximately standard normally distributed? If yes, why, if no, what needs to happen to make it normally distributed?

Answer 24

No, it needs the parent distribution to be normal, or to have a large enough population (n>30) to be approximately normal.

Question 25

What is a t-curve and how does it compare to a standard normal distribution?

Answer 25

A t-curve is the density curve of a t-distribution.

-Looks similar but has a wider spread (longer tails)
-Both symmetric about 0

Question 26

What is the only parameter that determines the shape of a t-curve?

Answer 26

It’s degrees of freedom which is = n-1

Question 27

Similar to a normal random variable, for a random variable T with a t-distribution, what do it’s probabilities equal?

Answer 27

The area under the variable’s associated t-curve

Remember the total area under the curve is equal to 1

Question 28

When the degrees of freedom become ______, t-curves look increasingly like the standard normal curve

Answer 28

larger

So basically the larger the sample size, the closer it looks to a standard normal curve

Question 29

Similar to table II, a table IV can be used to find some t-scores. What is the major way in which they differ?

Answer 29

The table cannot be used to find the probabilities directly

(With table II the numbers in the middle of the table equal the probabilities, it does not do that with the table IV)

Question 30

How do we use the table IV to find the t-score?

Answer 30

First, determine the degree of freedom (df=n-1) and then find this number along the columns on the left and right which will tell us which row to look at

At the top are the t0.1, t0.05 etc. which tell us which column to look at

The desired t-score is at the intersection

Question 31

What should we do if the exact degree of freedom is not in the table? (i.e. 99)

Answer 31

Go to the next smaller value

Question 32

If x̄ is approximately normally distributed and σ is unknown, how do we determine the confidence interval for the population mean?

Answer 32

x̄ +/- (tα/2) x (s/√n)

Question 33

What does tα/2 represent?

Answer 33

The t-score for t-distribution with
df = n-1

Question 34

Determine the t-score when looking at a sample size of 11 and t0.025

Answer 34

2.228

Question 35

What is the total area under a t-curve equal to?

Answer 35

1

similar to normal random variables using a z value

Question 36

For a random variable with a T distribution, what are it’s properties equal to?

Answer 36

The areas under the associated t-curve

similar to normal random variables using a z value