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Introduction to Inference

Introduction to Inference
The purpose of inference is to draw conclusions from data. Conclusions take into account the natural variability in the data, therefore formal inference relies on probability to describe chance variation. We will go over the two most prominent types of formal statistical inference Confidence Intervals for estimating the value of a population parameter. Tests of significance which asses the evidence for a claim. Both types of inference are based on the sampling distribution of statistics. That is both report probabilities that state what would happen if we used the inference method many times.

Introduction to Inference
Since both methods of formal inference are based on sampling distributions, they require probability model for the data. The model is most secure and inference is most reliable when the data are produced by a properly randomized design. When we use statistical inference we assume that the data come from a randomly selected sample or a randomized experiment.

Estimating with Confidence
Community banks are banks with less than a billion dollars of assets. There are approximately 7500 such banks in the United States. In many studies of the industry these banks are considered separately from banks that have more than a billion dollars of assets. The latter banks are called “large institutions.” The community bankers Council of the American bankers Association (ABA) conducts an annual survey of community banks. For the 110 banks that make up the sample in a recent survey, the mean assets are = 220 (in millions of dollars). What can we say about , the mean assets of all community banks?

Estimating with Confidence
The sample mean is the natural estimator of the unknown population mean . We know that is an unbiased estimator of . The law of large numbers says that the sample mean must approach the population mean as the size of the sample grows. Therefore, the value = 220 appears to be a reasonable estimate of the mean assets  for all community banks. What if we want to do more than just provide a point estimate? second sample would surely not give 220 again. Unbiased ness says only that there is no systematic tendency to underestimate or overestimate the truth. Could we plausibly get a sample mean of 250 or 200 on repeated samples? An estimate without an indication of its variability is of limited value.

Estimating with Confidence
Suppose that we are interested in the value of some parameter (for example ), and we want to construct a confidence interval for it, specifying some particular desired level of confidence.

Estimating with Confidence
If we have a way to estimate this parameter from sample data (using an estimator, for example sample mean), and we know the distribution of the estimator, we can use this knowledge to construct a probability statement involving both the estimator and the true value of the parameter which we are trying to estimate. This statement is manipulated mathematically to yield confidence limits.

Confidence Interval A level C confidence interval for a parameter has the following form: An interval calculated from the data, usually of the form Estimate  [Factor][SE of estimate] The value of the factor will depend upon the level of confidence desired, and the distribution of the estimator.

Confidence Interval Suppose we are investigating a continuous random variable x, which is normally distributed with a mean  and variance 2. We can estimate the population mean  using the sample mean , calculated from a random sample of n observations; 2 can also be estimated using the sample variance S2.

Confidence Interval We can also estimate the standard error of the sample mean,

Confidence Interval The value of the factor will depend upon the level of confidence desired, and the distribution of the estimator. The sampling distribution is exactly N(, ) when the population has the N(, ) distribution. The central Limit theorem says that this same sampling distribution is approximately correct for large samples whenever the population mean and standard deviation are  and .

Confidence Interval for a Population Mean
To construct a level C confidence interval, 1st catch the central C area under a Normal curve. Since all Normal distributions are the same in the standard scale, we obtain what we need from the standard Normal curve.

Confidence Interval for a Population Mean
The figure in previous slide shows the relationship between central area C and the points z* that marks off this area. Values of z* for many choices of C can be found from standard Normal table (table A). Here are some examples; Z* C 90% 95% 99%

Confidence Interval for a Population Mean
Any Normal curve has probability C between the points z* standard deviation below the mean and the point z* above the mean. Sample mean has Normal distribution with mean  and standard deviation n.

Confidence Interval for a Population Mean
So there is probability C that lies between This is exactly the same as saying that the unknown population mean  lies between

Confidence Interval for a Population Mean
Choose a SRS of size n from a population having unknown mean  and known standard deviation . A level C confidence interval for  is Here z* is the critical value with area C between –z* and z* under the standard Normal curve. The quantity is the margin of error. The interval is exact when the population distribution is normal and is approximately correct when n is large in other cases.

Example: Banks’ loan –to-deposit ration
The ABA survey of community banks also asked about the loan-to-deposit ratio (LTDR), a bank’s total loans as a percent of its total deposits. The mean LTDR for the 110 banks in the sample is and the standard deviation is s = This sample is sufficiently large for us to use s as the population  here. Find a 95% confidence interval for the mean LTDR for community banks.

How Confidence Intervals behave?
The margin of error z*n for estimating the mean of a Normal population illustrate several important properties that are shared by all confidence intervals in common use. Higher confidence level increases z* and therefore increases the margin of error for intervals based on the same data. If the margin of error is too large, there are two ways to reduce it: Use a lower level confidence (smaller c, hence smaller z*) Increase the sample size (larger n)

Example: Banks’ loan –to-deposit ration
Suppose there were only 25 banks in the survey of community banks, and that X and  are unchanged. The margin of error increases from 2.3 to A 95% confidence interval for  is:

Example: Banks’ loan –to-deposit ration
Suppose that we demand 99% confidence interval for the mean LTDR rather than 95% when n is 110. The margin of error increases from 2.3 to What is the 99% confidence interval?

Choosing Sample size. A wise user of statistics never plans data collection without at the same time planning the inference. One can arrange to have both high confidence and a small margin of error. To obtain the desired margin of error m, set the expression equal to m, substitute the critical value z* for your desired confidence level, and solve for the sample size n.

Choosing Sample size. The confidence interval for a population mean will have a specified margin of error m when the sample size is

How many banks should we survey?
In our previous example we found that the margin of error was 2.3 for estimating the mean LTDR of community banks with n = 110 and 95% confidence. We are willing to settle for a margin of error of 3.0 when we do the survey next year. How many banks should we survey if we still want 95% confidence interval?

Tests of Significance Confidence intervals are appropriate when our goal is to estimate a population parameter. The second type of inference is directed at assessing the evidence provided by the data in favor of some claim about the population. A significance test is a formal procedure for comparing observed data with a hypothesis whose truth we want to assess. The hypothesis is a statement about the parameters in a population or model. The results of a test are expressed in terms of a probability that measures how well the data and the hypothesis agree.

Example: Bank’s net income
The community bank survey described in previous lecture also asked about net income and reported the percent change in net income between the first half of last year and the first half of this year. The mean change for the 110 banks in the sample is Because the sample size is large, we are willing to use the sample standard deviation s = 26.4% as if it were the population standard deviation . The large sample size also makes it reasonable to assume that is approximately normal.

Example: Bank’s net income
Is the 8.1% mean increase in a sample good evidence that the net income for all banks has changed? The sample result might happen just by chance even if the true mean change for all banks is  = 0%. To answer this question we asks another Suppose that the truth about the population is that = 0% (this is our hypothesis) What is the probability of observing a sample mean at least as far from zero as 8.1%?

Example: Bank’s net income
The answer is: Because this probability is so small, we see that the sample mean is incompatible with a population mean of  = 0. We conclude that the income of community banks has changed since last year.

Example: Bank’s net income
We calculated a probability assuming that the true mean is zero ( = 0 ). This probability guides our final choice. If the probability is very small, the data don’t fit our assumption that ( = 0 ) and we conclude that the mean is not in fact zero. The fact that the calculated probability is very small leads us to conclude that the average percent change in income is not in fact zero.

Example:Is this percent change different from zero?
Suppose that next year the percent change in net income for a sample of 110 banks is = 3.5%. (We assume that the standard deviation  = 26.4%.) This sample mean is closer to the value  = 0 corresponding to no mean change in income. What is the probability that the mean of a sample of size n = 110 from a normal population with  = 0 and standard deviation  = 26.4 is as far away or farther away from zero as ?

Example:Is this percent change different from zero?
The answer is: A sample result this far from zero would happen just by chance in 8% of samples from a population having true mean zero. An outcome that could so easily happen by chance is not good evidence that the population mean is different from from zero.

Example:Is this percent change different from zero?
The mean change in net assets for a sample of 110 banks will have this sampling distribution if the mean for the population of all banks is  = 0. A sample mean could easily happen by chance. A sample mean is far out on the curve that it would rarely happen just by chance.

Tests of Significance: Formal details
The first step in a test of significance is to state a claim that we will try to find evidence against. Null Hypothesis H0 The statement being tested in a test of significance is called the null hypothesis. The test of significance is designed to assess the strength of the evidence against the null hypothesis. Usually the null hypothesis is a statement of “no effect” or “no difference.” We abbreviate “null hypothesis” as H0.

Tests of Significance: Formal details
A null hypothesis is a statement about a population, expressed in terms of some parameter or parameters. The null hypothesis in our bank survey example is H0 :  = 0 It is convenient also to give a name to the statement we hope or suspect is true instead of H0. This is called the alternative hypothesis and is abbreviated as Ha. In our bank survey example the alternative hypothesis states that the percent change in net income is not zero. We write this as Ha :   0

Tests of Significance: Formal details
Since Ha expresses the effect that we hope to find evidence for we often begin with Ha and then set up H0 as the statement that the Hoped-for effect is not present. Stating Ha is not always straight forward. It is not always clear whether Ha should be one-sided or two-sided. The alternative Ha :   0 in the bank net income example is two-sided. In any given year, income may increase or decrease, so we include both possibilities in the alternative hypothesis.

Example:Have we reduced processing time?
Your company hopes to reduce the mean time  required to process customer orders. At present, this mean is 3.8 days. You study the process and eliminate some unnecessary steps. Did you succeed in decreasing the average process time? You hope to show that the mean is now less than 3.8 days, so the alternative hypothesis is one sided, Ha :  < 3.8. The null hypothesis is as usual the “no change” value, H0 :  = 3.8.

Tests of Significance: Formal details
Test statistics We will learn the form of significance tests in a number of common situations. Here are some principles that apply to most tests and that help in understanding the form of tests: The test is based on a statistic that estimate the parameter appearing in the hypotheses. Values of the estimate far from the parameter value specified by H0 gives evidence against H0. We will learn the form of significance tests in a number of common situations. Here are some principles that apply to most tests and that help in understanding the form of tests:

Tests of Significance: Formal details
A test statistic measures compatibility between the null hypothesis and the data. Many test statistics can be thought of as a distance between a sample estimate of a parameter and the value of the parameter specified by the null hypothesis.

Example: bank’s income
The hypotheses: H0 :  = 0 Ha :   0 The estimate of  is the sample mean Because Ha is two-sided, large positive and negative values of (large increases and decreases of net income in the sample) counts as evidence against the null hypothesis.

Example: bank’s income
The test statistic The null hypothesis is H0 :  = 0, and a sample gave the The test statistic for this problem is the standardized version of : This statistic is the distance between the sample mean and the hypothesized population mean in the standard scale of z-scores.

Tests of Significance: Formal details
The test of significance assesses the evidence against the null hypothesis and provides a numerical summary of this evidence in terms of probability. P-value The probability, computed assuming that H0 is true, that the test statistic would take a value extreme or more extreme than that actually observed is called the P-value of the test. The smaller the p-value, the stronger the evidence against H0 provided by the data. To calculate the P-value, we must use the sampling distribution of the test statistic. Extreme means far from what we would expect if H0 were true

Example: bank’s income
The P-value In our banking example we found that the test statistic for testing H0 :  = 0 versus Ha :   0 is If the null hypothesis is true, we expect z to take a value not far from 0. Because the alternative is two-sided, values of z far from 0 in either direction count ass evidence against H0. So the P-value is:

Example: bank’s income
The p-value for bank’s income. The two-sided p-value is the probability (when H0 is true) that takes a value at least as far from 0 as the actually observed value.

Tests of Significance: Formal details
We know that smaller P-values indicate stronger evidence against the null hypothesis. But how strong is strong evidence? One approach is to announce in advance how much evidence against H0 we will require to reject H0. We compare the P-value with a level that says “this evidence is strong enough.” The decisive level is called the significance level. It is denoted be the Greek letter .

Tests of Significance: Formal details
If we choose  = 0.05, we are requiring that the data give evidence against H0 so strong that it would happen no more than 5% of the time (1 in 20) when H0 is true. Statistical significance If the p-value is as small or smaller than , we say that the data are statistically significant at level .

Tests of Significance: Formal details
You need not actually find the p-value to asses significance at a fixed level . You can compare the observed test statistic z with a critical value that marks off area  in one or both tails of the standard Normal curve.

Test for a Population Mean

Two Types of Error In tests of hypothesis, there are simply two hypotheses, and we must accept one and reject the other. We hope that our decision will be correct, but sometimes it will be wrong. There are two types of incorrect decisions. If we reject H0 when in fact H0 is true. This is called type I error. If we accept H0 when in fact Ha is true. This is called Type II error.

Two Types of Error

Two Types of Error Significance and type I error
The significance level  of any fixed level test is the probability of a type I error. That is,  is the probability that the test will reject the null hypothesis H0 when in fact H0 is true.