This means our point estimator should be an and have a sampling distribution with.

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Prudence Caldwell
5 years ago
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1 Suppose we have a Normal population with an unknown mean, μ, and standard deviation of 36. In groups, I would like you to estimate the mean, μ, of this population. I will generate a sample of size 16 from this population and since the sample mean,, is an unbiased estimator of population mean, I will give you the mean from this one sample,. Your job is to come up with a method to Dind an interval of REASONABLE values that will contain the true value μ. De#inition: A point estimator is a STATISTIC that provides an estimate of a population parameter. The value of that statistic from a sample is called a point estimate. Ideally, a point estimate is our "best guess" at the value of an unknown parameter. This means our point estimator should be an and have a sampling distribution with. 1

2 So we have a point estimator,, of our unknown parameter - μ - and we want to know how close this estimate is to the actual μ. It is VERY unlikely that we got the actual value of μ with our point estimator,. In order to answer this we need to know how the sample mean,, would vary if we took many SRSs of the same size (16 in this case) from this same population. SAMPLING DISTRIBUTIONS answer this question. Sampling Distribution of : Shape: Because we have a Normal distribution to start with our sampling distribution of is also Normal. Center: The mean of the sampling distribution of μ (the parameter we are trying to estimate) is the same as Spread: The standard deviation of the sampling distribution of for an SRS of n = 16 is 9 = 5 because the 10% condition is met. 2

Sampling Distribution of Population Distribution = 5 9 σ = 20 36 μ Values of X μ Values of 1. To estimate μ we use the sample mean,, of the random sample.

3 Sampling Distribution of Population Distribution = 5 9 σ = μ Values of X μ Values of 1. To estimate μ we use the sample mean,, of the random sample. We don't expect to be exactly equal to μ, so we want to say how accurate the estimate is. 2. In repeated samples, the values of follow a Normal distribution with mean μ and standard deviation, as in the Digure above. 3. The 95 part of the rule for Normal distributions says that is within 2 standard deviations of the population mean μ in about 95% of all samples of size n. 95% of all sample s µ - 2 µ µ Whenever is within 2 of μ, μ is within 2 of. This happens in about 95% of all possible samples. So the interval from μ - 2 to μ + 2 "captures" the population mean μ in about 95% of all samples of size n. 5. If we estimate that μ lies somewhere between μ - 2 and μ + 2, we'd be calculating this interval using a method that captures the true μ in about 95% of all possible samples of this size. 3

4 All condidence intervals will have a form similar to this: estimate ± margin of error Keep in mind this "error" has to do with chance variation due to random sampling or random assignment - i.e. sampling variability - not whether or not we made an error in calculating or ANY other type of error. De#initions: A con2idence interval for a parameter has two parts: The margin of error tells how close the estimate tends to be to the unknown parameter in repeated random sampling. A con2idence level, C, which gives the overall success rate of the METHOD for calculating the condidence interval. That is in C% of ALL POSSIBLE SAMPLES, the method would yield an interval that captures the true parameter value. 4

5 Interpreting Con#idence Levels and Con#idence Intervals 1. Go to and launch the applet (Con$idence Interval applet). The default setting for the condidence level is 95%. Change this to 90%. We will be using this applet to investigate the idea of condidence level. 2. Click "Sample" to choose an SRS and display the resulting condidence interval. Did the interval capture the population mean μ (what the applet calls a "hit")? Do this a total of 10 times. How many of the intervals captured the population mean μ? 3. Reset the applet. Click "Sample 50" to choose 50 SRSs and display the condidence intervals based on those samples. How many captured the parameter μ? 4. Keep clicking "Sample 50" and observe the value of "Percent hit". What do you notice? As the applet condirmed, the con2idence level is the overall "capture rate" if the method is used many times. A 95% condidence level implies that in repeated sampling, 95% of the intervals constructed in this way will contain the unknown population mean μ. 5

6 Con2idence LEVEL: To say that we are 95% con$ident is shorthand for "95% of all possible samples of a given size from this population will result in an interval that captures the unknown parameter." Con2idence INTERVAL: To interpret a C% condidence interval for an unknown parameter, we say, "We are C% condident that the interval from to contains the actual value of the [population parameter IN CONTEXT]." What's the probability that our 95% con2idence interval captures the parameter? NOT 95%!!! This level only talks about the condidence in the METHOD, not ONE sample! BEFORE we sample, we have a 95% chance of getting a sample that will produce a 95% condidence interval that captures the true parameter. However, once we sample our resulting condidence interval either will contain the parameter or won't - i.e. the probability that it contains the parameter is either 1 (it did) or 0 (it didn't). The con2idence level DOES NOT tell us the chance that A PARTICULAR con2idence interval contains the population parameter. 6

7 Something to keep in mind: A condidence interval is a statement about a parameter - not a sample statistic. DON'T say "We are 95% condident that the interval from to contains the sample proportion/ mean of all " We know it contains the sample statistic so this makes NO sense! Check your understanding: How much does the fat content of Brand X hot dogs vary? To Dind out, researchers measured the fat content (in grams) of a random sample of 10 Brand X hot dogs. A 95% condidence interval for the population standard deviation σ is 2.84 to Interpret this condidence interval. 2. Interpret the condidence level. 3. True or false: The interval from 2.84 to 7.55 has a 95% chance of containing the actual population standard deviation σ. Justify your answer. 7

8 Choosing a con#idence level Return to the Con$idence Interval applet. We are going to explore the relationship between the condidence level and the condidence interval. 1. Set the condidence level at 95% and click "Sample 50". Observe the percent hit and the length of the condidence interval. 2. Change the condidence level to 99%. What happens to the length of the condidence interval? To the percent hit? 3. Now change the condidence level to 90%. What happens to the length of the condidence interval? To the percent hit? 4. Finally, change the condidence level to 80%. What happens to the length of the condidence interval? To the percent hit? As we increase condidence level, we increase the width of our interval... What else might effect the width of our condidence interval? Let's look at the form of the condidence interval more closely: estimate ± (critical value)(standard deviation of the statistic) Clearly the smaller the margin of error, the narrower the condidence interval - the more precise our estimate of the population parameter. 8

9 The margin of error depends on two things: The critical value - this depends on the sampling distribution (more on this) and on the condidence level - the greater the condidence the greater the critical value The standard deviation of the statistic - this depends on the sample size n - the bigger the sample, the smaller the standard deviation of the statistic, and the narrower the interval. 9

10 CONDITIONS: 1. Random: The data should come from a well-designed random sample or a randomized experiment. 2. Normal: The methods we use to construct condidence intervals for μ and p depend on the fact that the sampling distribution of the statistic ( or ) is at least approximately Normal For means: For proportions: 3. Independent: The procedures for calculating condidence intervals assume that individual observations are independent. Random sampling helps ensure this so long as our population is indinite. If NOT and we are sampling without replacement we should check the 10% condition. Homework: p. 481 #s 1-17 odd, all 10

MITOCW ocw f99-lec30_300k

MITOCW ocw f99-lec30_300k MITOCW ocw-18.06-f99-lec30_300k OK, this is the lecture on linear transformations. Actually, linear algebra courses used to begin with this lecture, so you could say I'm beginning this course again by