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Statistics and Quantitative Analysis U4320

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Statistics and Quantitative Analysis U4320

Segment 6

Prof. Sharyn O’Halloran

- 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.

- 1.Samples

- 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_.

- 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

- Definition

- 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.

- 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

- 1. Motivation

- 2. Constructing a 95% Confidence Interval
- a. Graph
- First, we know that the sample mean is distributed normally, with mean_ and standard error

- a. Graph

- 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.

- 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:

- 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

- What's the probability that the population mean will fall within the interval 1.96 * SE?

- 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:

- For a 95% confidence interval:

- 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.

- Answer:
- Why 95%?

- 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

- 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, _.

- 90% of the time, the mean will lie with in this range.

- 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.

- 1.Characteristics of a Student-t distribution

- b. When to use t-distribution
- i.s is unknown
- ii.Sample size n is small

- 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.

- 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

- Answer:

- 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:

- This would be the formula if the sample size were large and we knew both 1 and 2.

- Example: Test scores of two classes where one is from an inner city school and the other is from an affluent suburb.

- 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:

- 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:

- The degrees of freedom are (n1-1) + (n2-1), or (n1+n2-2).

- Two classes from the same school take a test. Calculate the 95% confidence interval for the difference between the two class means.

- Answer:

- 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.

- 1. Definition
- 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.

- 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:

- 4. Example:

- Notice that the standard error here is much smaller than in most of our unmatched pair examples.

- 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:

- 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%.