Estimating with Confidence:

 

Suppose I want to know how often teenagers go to the movies.  Specifically, I want to know how many times per month a typical teenager (ages 13 through 17) goes to the movies.

 

Suppose I take an SRS of 100 teenagers and calculate the sample mean to be .

 

The sample mean is an unbiased estimator of the unknown population mean , so I would estimate the population mean to be approximately 2.1.  However, a different sample would have given a different sample mean, so I must consider the amount of variation in the sampling model for .

 

         The sampling model for  is approximately normal.

         The mean of the sampling model is .

         The standard deviation of the sampling model is  assuming the population size is at least 10n.

                                                           

 

Suppose we know that the population standard deviation is .  Then the standard deviation for the sampling model is

 

 

Then 95% of our samples will produce a statistic  that is between  and .

 

Therefore in 95% of our samples, the interval between  and  will contain the parameter .

 

 

The margin of error is 0.10.

 

 

For our sample of 100 teenagers, .  Because the margin of error is 0.10, then we are 95% confident that the true population mean lies somewhere in the interval , or [2.0, 2.2].

 

The interval [2.0, 2.2] is a 95% confidence interval because we are 95% confident that the unknown  lies between 2.0 and 2.2.

 

Answers to Exercises:

#10.1

a)         44% to 50%

b)         The statistic 47% came from a small sample; it is a good estimate, but we do not expect it to be exactly correct.

c)         In 95% of all samples, the true value will be within 3% of the statistic.

 

#10.2

a)         The mean of the sampling model is 280 and the standard deviation is , assuming the population is at least 10,000 people.

 

b)                                                                                             N(280, 1.9)

                          274.3  276.2  278.1  280  281.9  283.8  285.7

 

 


c)         2 standard deviations, or 3.8 points

e)         95%

 

#10.3

NO!  95% is not a probability or a proportion; it is a confidence level.  We are 95% confident that the unknown parameter   lies between 267.8 and 276.2.  Also, the parameter is a mean, not an individual score.

 

 

 

Start with sample data.  Compute an interval that has probability C of containing the true value of the parameter.  This is called a level C confidence interval.

 

How do we construct confidence intervals?

Since the sampling model of the sample mean  is approximately normal, we can use normal calculations to construct confidence intervals.

 

For a 95% confidence interval, we want the interval corresponding to the middle 95% of the normal curve.

For a 90% confidence interval, we want the interval corresponding to the middle 90% of the normal curve.

And so on…

 

If we are using the standard normal curve, we want to find the interval using z-values.

 

 

 

 

Suppose we want to find a 90% confidence interval for a standard normal curve.  If the middle 90% lies within our interval, then the remaining 10% lies outside our interval.  Because the curve is symmetric, there is 5% below the interval and 5% above the interval.  Find the z-values with area 5% below and 5% above.

 

These z-values are denoted .  Because they come from the standard normal curve, they are centered at mean 0. 

 is called the upper p critical value, with probability p lying to its right under the standard normal curve.

 

To find p, we find the complement of C and divide it in half, or find .

 

For a 95% confidence interval, we want the z-values with upper p critical value 2.5%.

For a 99% confidence interval, we want the z-values with upper p critical value 0.5%.

 

Remember that z-values tell us how many standard deviations we are above or below the mean.

 

To construct a 95% confidence interval, we want to find the values 1.96 standard deviation below the mean and 1.96 standard deviations above the mean, or .

Using our sample data, this is , assuming the population is at least 10n.

 

In general, to construct a level C confidence interval using our sample data, we want to find .

 

The estimate for  is .

The margin of error is .  Note that the margin of error is a positive number.  It is not an interval.

 

 

We would like high confidence and a small margin of error.

A higher confidence level means a higher percentage of all samples produce a statistic close to the true value of the parameter.  Therefore we want a high level of confidence.

A smaller margin of error allows us to get closer to the true value of the parameter, so we want a small margin of error.

 

So how do we reduce the margin of error?

         Lower the confidence level (by decreasing the value of z*)

         Lower the standard deviation

         Increase the sample size.  To cut the margin of error in half, increase the sample size by four times the previous size.

 

You can have high confidence and a small margin of error if you choose the right sample size.

 

 

 

To determine the sample size n that will yield a confidence interval for a population mean with a specified margin of error m, set the expression for the margin of error to be less than or equal to m and solve for n.

 

CAUTION!!

These methods only apply to certain situations.  In order to construct a level C confidence interval using the formula , for example, the data must be an SRS and we must know the population standard deviation.  Also, we want to eliminate (if possible) any outliers.

 

The margin of error only covers random sampling errors.  Things like undercoverage, nonresponse, and poor sampling designs can cause additional errors.

 

 

 

Tests of Significance:

 

Example 10.7

Diet colas use artificial sweeteners to avoid sugar.  Colas with artificial sweeteners gradually lose sweetness over time.  Manufacturers therefore test new colas for loss of sweetness before marketing them.  Trained tasters sip the cola along with drinks of standard sweetness and score the cola on a “sweetness score” of 1 to 10.  The cola is then stored for a period of time, then each taster scores the stored cola.  This is a matched pairs experiment.  The reported data is the difference in tasters’ scores.  The bigger the difference, the bigger the loss in sweetness.

 

                        2.0                   0.4                   0.7                   2.0                   -0.4

                        2.2                   -1.3                  1.2                   1.1                   2.3

 

The sample mean  indicates a small loss of sweetness.

 

Consider that a different sample of tasters would have resulted in different scores, and that some variation in scores is expected due to chance.

 

Does the data provide good evidence that the cola lost sweetness in storage?

 

To answer that question, we will perform a significance test.

 

1.         Identify the parameter.

            The parameter of interest is , the mean loss in sweetness.

 

2.         State the null hypothesis.

            There is no effect or change in the population.  This is the statement we are trying to find evidence against.

            The cola does not lose sweetness.

           

 

            State the alternative hypothesis.

            There is an effect or change in the population.

            This is the statement we are trying to find evidence for.

The cola does lose sweetness.

           

 

3.         Calculate a statistic to estimate the parameter.

            Is the value of the statistic far from the value of the parameter?  If so, reject the null hypothesis.  If not, accept the null hypothesis.

 

4.         Calculate the P-value. 

If it is small, your result is statistically significant.

Suppose the individual tasters’ scores vary according to a normal distribution with mean  and .  We want to test the null hypothesis so we assume .

 

 

So the sampling model for  is approximately normal with mean  and standard deviation

N(0, 0.316)

 

 

 

 

 

 

 

 

 

          -0.9    -0.6    -0.3     0       0.3    0.6    0.9

 

Our sample mean, , was 1.02.  Assuming that the null hypothesis is true, what is the probability of getting a result at least that large? 

 

Normalcdf ( 1.02, 1E99, 0, 0.316 ) = 0.0006

The probability to the right of  is called the P-value.

 

The P-value is 0.0006, meaning that we would only expect to get this result in 6 out of 10,000 samples.  This is very unlikely, so we will reject the null hypothesis in favor of the alternative hypothesis and conclude that the cola actually did lose sweetness.

 

If the P-value is small we say that our result is statistically significant.  The smaller the P-value, the stronger the evidence provided by the data.

 

How small is small enough?  Compare the P-value to the value of the significance level This value is usually predetermined.

 

If the P-value is as small or smaller than , we say that the data are statistically significant at level .

 

 

Hypotheses can be one-sided or two-sided.

          is a one-sided hypothesis because we are only looking at one direction, greater than.

          is a two-sided hypothesis because we are looking at two directions, greater than and less than.

 

 

 

 

 

Inference as Design:

 

If we use the results of a significance test to make a decision, then we either reject the null hypothesis in favor of the alternative hypothesis, or we accept the null hypothesis.  This is called acceptance sampling.

 

We hope that our decision will be correct, but it is possible that we make the wrong decision.  There are two ways to make a wrong decision:

¯                            We can reject the null hypothesis when in fact it is true.  This is called a Type I Error.

¯                            We can accept (fail to reject) the null hypothesis when in fact it is false.  This is called a Type II Error.

 

 

 

Truth about the population

 

 

H0 is true

H0 is false

Decision based on sample

Reject H0

Type I Error

p = 

Correct Decision

p = 1 – 

Accept H0

Correct Decision

Type II Error

p = 

 

We are interested in knowing the probability of making a Type I Error and the probability of making a Type II Error.

 

 

A Type I Error occurs if we reject the null hypothesis when it is in fact true.  When do we reject the null hypothesis?  When we assume that it is true and find that the statistic of interest falls in the rejection region.  The probability that the statistic falls in the rejection region is the area of the shaded region, or .

 

                                                      One-Sided Test

 

 

 

 

 

 

 

 

 

Therefore the probability of a Type I Error is equal to the significance level of a fixed level test.  The probability that the test will reject the null hypothesis H0 when in fact H0 is true is .

 

 

A Type II Error occurs if we accept (or fail to reject) the null hypothesis when it is in fact false.  When do we accept (or fail to reject) the null hypothesis?  When we assume that it is true and find that the statistic of interest falls outside the rejection region. 

 

However, the probability that the statistic falls outside the rejection region is NOT the area of the unshaded region.  Think about it… If the null hypothesis is in fact false, then the picture is NOT CORRECT… it is off center.

 

                                                        Two-Sided Test

 

 

 

 

 

 

 

 

 

To calculate the probability of a Type II Error, we must find the probability that the statistic falls outside the rejection region (the unshaded area) given that the mean is some other specified value.

 

 

 

 

 

 

 

 

 

 

 

 


                      lower                        upper

                     critical                       critical

                      value                        value

 

 

 

 

 

 

 

 

 

 

 


                      lower                        upper

                     critical                       critical

                      value                        value

The probability of a Type II Error tells us the probability of accepting the null hypothesis when it is actually false.

 

The complement of this would be the probability of not accepting (in other words rejecting) the null hypothesis when it is actually false.

 

To calculate the probability of rejecting the null hypothesis when it is actually false, compute            1 – P(Type II Error), or (1 – .  This is called the power of a significance test.