Statistics and quantitative analysis u4320
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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.

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Statistics and Quantitative Analysis U4320

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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

    • Answer:

    • Why 95%?


    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

    • Answer:


    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

    • Answer:


    II. Confidence Interval


    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.

    • 2. Advantage

      • 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

    • 4. Example:


    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.

      • Answer: n= 1500

      • Sample proportion P:


    IV. Confidence Intervals for Proportions

    • Create a 95% confidence interval:

    • where  and P are the population and sample proportions, and n is the sample size.

    • That is, with 95% confidence, the proportion for Clinton in the whole population of voters was between 53% and 59%.


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