Statistics and Quantitative Analysis U4320

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# Statistics and Quantitative Analysis U4320 - PowerPoint PPT Presentation

Statistics and Quantitative Analysis U4320. Segment 6 Prof. Sharyn O’Halloran. I. Introduction. A. Review of Population and Sample Estimates. B. Sampling 1. Samples The sample mean is a random variable with a normal distribution.

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### Statistics and Quantitative Analysis U4320

Segment 6

Prof. Sharyn O’Halloran

I. Introduction
• A. Review of Population and Sample Estimates
• B. Sampling
• 1. Samples
• The sample mean is a random variable with a normal distribution.
• If you select many samples of a certain size then on average you will probably get close to the true population mean.
I. Introduction
• 2. Central Limit Theorem
• Definition
• If a Random Sample is taken from any population with mean_ and standard deviation_, then the sampling distribution of the sample means will be normally distributed with
• 1) Sample Mean E() = _, and
• 2) Standard Error SE() = _/_n
• As n increases, the sampling distribution of tends toward the true population mean_.
I. Introduction
• C. Making Inferences
• To make inferences about the population from a given sample, we have to make one correction, instead of dividing by the standard deviation, we divided by the standard error of the sampling process:
• Today, we want to develop a tool to determine how confident we are that our estimates lie within a certain range.
II. Confidence Interval
• A. Definition of 95% Confidence Interval ( known)
• 1. Motivation
• We know that, on average, is equal to_.
• We want some way to express how confident we are that a given is near the actual_ of the population.
• We do this by constructing a confidence interval, which is some range around that most probably contains_.
• The standard error is a measure of how much error there is in the sampling process. So we can say that is equal to__ the standard error
•  = X_Standard Error
II. Confidence Interval
• 2. Constructing a 95% Confidence Interval
• a. Graph
• First, we know that the sample mean is distributed normally, with mean_ and standard error
II. Confidence Interval
• b. Second, we determine how confident we want to be in our estimate of_.
• Defining how confident you want to be is called the -level.
• So a 95% confidence interval has an associate - level of .05.
• Because we are concerned with both higher and lower values, the relevant range is _/2 probability in each tail.
II. Confidence Interval
• C. Define a 95% Confidence Range
• Now let's take an interval around  that contains 95% of the area under the curve.
• So if we take a random sample of size n from the population, 95% of the time the population mean  will be within the range:
• What is a z value associated with a .025 probability?
• From the z-table, we find the z-value associated with a .025 probability is 1.96.
• So our range will be bounded by:
II. Confidence Interval
• Now, let's take this interval of size [-1.96 * SE , 1.96 * SE] and use it as a measuring rod.
• d. Interpreting Confidence Intervals
II. Confidence Interval
• What's the probability that the population mean  will fall within the interval  1.96 * SE?
• e. In General
• We get the actual interval as 1.96 * SE on either side of the sample mean .
• We then know that 95% of the time, this interval will contain . This interval is defined by:
II. Confidence Interval
• For a 95% confidence interval:
• Examples
• 1. Example 1: Calculate a 95% confidence Interval
• Say we sample n=180 people and see how many times they ate at a fast-food restaurant in a given week. The sample had a mean of 0.82 and the population standard deviation  is 0.48. Calculate the 95% confidence interval for these data.
II. Confidence Interval
• 2. Example 2: Calculating a 90% confidence interval
• A random sample of 16 observations was drawn from a normal population with s = 6 and = 25. Find a 90% (a = .10) confidence interval for the population mean, _.
• First, find Z.10/2 in the standard normal tables
• Second, calculate the 90% confidence interval
II. Confidence Interval
• 90% of the time, the mean will lie with in this range.
• What if we wanted to be 99% of the time sure that the mean falls with in the interval?
• What happens when we move from a 90% to a 99% confidence interval?
II. Confidence Interval
• C. Confidence Intervals (_ unknown)
• 1.Characteristics of a Student-t distribution
• a.Shape the student t-distribution
• The t-distribution changes shape as the sample size gets larger, and in the limit it becomes identical to the normal.
II. Confidence Interval
• b. When to use t-distribution
• i. s is unknown
• ii. Sample size n is small
• 2. Constructing Confidence Intervals Using t-Distribution
• A. Confidence Interval
• 95% confidence interval is:
• B. Using t-tables
• Say our sample size is n and we want to know what's the cutoff value to get 95% of the area under the curve.
II. Confidence Interval
• i) Find Degrees of Freedom
• Degree of freedom is the amount of information used to calculate the standard deviation, s. We denote it as d.f. _ d.f. = n-1
• ii) Look up in the t-table
• Now we go down the side of the table to the degrees of freedom and across to the appropriate t-value.
• That's the cutoff value that gives you area of .025 in each tail, leaving 95% under the middle of the curve.
• iii) Example:
• Suppose we have sample size n=15 and t.025 What is the critical value? 2.13
II. Confidence Interval
• III. Differences of Means
• A. Population Variance Known
• Now we are interested in estimating the value (1 - 2) by the sample means, using 1 - 2.
• Say we take samples of the size n1 and n2 from the two populations. And we want to estimate the differences in two population means.
• To tell how accurate these estimates are, we can construct the familiar confidence interval around their difference:
II. Confidence Interval
• This would be the formula if the sample size were large and we knew both 1 and 2.
• B. Population Variance Unknown
• 1. If, as usual, we do not know 1 and 2, then we use the sample standard deviations instead. When the variances of populations are not equal (s1 s2):
• Example: Test scores of two classes where one is from an inner city school and the other is from an affluent suburb.
II. Confidence Interval
• 2. Pooled Sample Variances, s1 = s2 (s2 is unknown)
• If both samples come from the same population (e.g., test scores for two classes in the same school), we can assume that they have the same population variance . Then the formula becomes:

or just

• In this case, we say that the sample variances are pooled. The formula for is:
II. Confidence Interval
• The degrees of freedom are (n1-1) + (n2-1), or (n1+n2-2).
• 3. Example:
• Two classes from the same school take a test. Calculate the 95% confidence interval for the difference between the two class means.
II. Confidence Interval
• C. Matched Samples
• 1. Definition
• Matched samples are ones where you take a single individual and measure him or her at two different points and then calculate the difference.
• One advantage of matched samples is that it reduces the variance because it allows the experimenter to control for many other variables which may influence the outcome.
II. Confidence Interval
• 3. Calculating a Confidence Interval
• Now for each individual we can calculate their difference D from one time to the next.
• We then use these D's as the data set to estimate , the population difference.
• The sample mean of the differences will be denoted .
• The standard error will just be:
• Use the t-distribution to construct 95% confidence interval:
II. Confidence Interval
• Notice that the standard error here is much smaller than in most of our unmatched pair examples.
IV. Confidence Intervals for Proportions
• Just before the 1996 presidential election, a Gallup poll of about 1500 voters showed 840 for Clinton and 660 for Dole. Calculate the 95% confidence interval for the population proportion  of Clinton supporters.