Unformatted text preview: Topic #3 Statistical Inference
Point Estimation Sampling Distribution of x o Sampling Distribution of p • Not on Exam 1 o H pothesis Testing Hy o Interval Estimation • Not on Exam 1
o o Form Your Team By End of 2nd Class! Slide 1 Slide 2 Statistical Inference
o Statistical Inference
o Definitions • A population is the set of all the elements of interest. • All 50,000 UCF students • A sample is a subset of the population. • This class Definitions (cont.) A parameter is a numerical characteristic of a population • usually unknown value • mean of a variable • Average GPA of all UCF students • An estimate is a statistical approximation of a parameter’s value • Average GPA in this class • If the estimate is computed from a sample, it is a sample statistic Slide 3 Slide 4 Statistical Inference
The purpose of statistical inference • obtain information about a population from information contained in a sample • The sample results provide only estimates of the values of the population characteristics o With proper sampling methods, the sample results proper the will will provide “good” estimates of the population characteristics. • See earlier coverage of different sampling methods
o o Point Estimation
In point estimation we use the data from the sample to compute a SINGLE VALUE of a sample statistic that serves as an ESTIMATE of a population parameter. o ESTIMATOR • Formula • Gives numerical value of parameter. • Example: formula for mean x=
n i We refer to x as the point estimator of the population mean μ. o s is the point estimator of the population standard deviation σ. o p is the point estimator of the population proportion p.
Slide 6 Slide 5 1 Point Estimation
o Sampling Error
The absolute value difference between an unbiased point estimate and the corresponding population parameter is called the sampling error. o Sampling error is the result of using a subset of the population (the sample), and not the entire population to develop estimates. o The sampling errors are: | x − μ | for sample mean |s - σ | for sample standard deviation | p − p | for sample proportion
o REVIEW • ESTIMATE • numerical value calculated from data in the sample • approximation of population parameter value • ESTIMATOR • Formula • Gives estimate Slide 7 Slide 8 Example: ESPN
ESPN annually receives 1,500 applications from prospective student interns. The application forms contain a variety of information including the individual’s scholastic aptitude test (SAT) score and whether or not the individual is an in-state resident. inThe director of interns would like to know, at least roughly, the following information: • the average SAT score for the applicants, and • the proportion of applicants that are in-state inresidents. We will now look at two alternatives for obtaining the desired information.
Slide 9 Example: ESPN
o Alternative #1: Take a Census of ALL 1,500 Applicants ALL • SAT Scores • Population Mean x μ = ∑ i = 990 1,500 • Population Standard Deviation (x − μ)2 = 80 σ= ∑ i 1,500 • In-State Applicants In• Population Proportion
p= 1,080 = .72 1,500 Slide 10 Example: Example: ESPN
o Example: ESPN
o Alternative #2: Take a SAMPLE of 50 Applicants SAMPLE • Excel can be used to select a simple random sample without replacement. • The process is based on random numbers generated by Excel’s RAND function. • RAND function generates numbers in the interval from from 0 to 1. • Any number in the interval is equally likely. • The numbers are actually values of a uniformly distributed random variable. Using Excel to Select a Simple Random Sample • 1500 random numbers are generated, one for each applicant in the population. • RULE FOR CHOOSING SAMPLE • then we choose the 50 applicants corresponding corresponding to the 50 smallest random numbers numbers as our sample. • (could have used different rule) • we find that 34 of the 50 are in-state residents in• Each of the 1500 applicants have the same probability of being included. Slide 11 Slide 12 12 2 Example: ESPN
o Example: Example: ESPN
o Point Estimates • x as Point Estimator of μ (μ = 990) x= • s as Point Estimator of σ (σ = 80) ∑ xi 49 , 850 = = 997 50 50
o Sampling error of… • …Mean = 997 – 990 = 7 • …Standard deviation = 75.2 – 80 = 4.8 in absolute value. • …Proportion = .68 - .72 = .04 in absolute value. s=
• 2 277 , 097 ∑ ( xi − x ) = = 75. 2 49 49 p as Point Estimator of p (p=.72) (p=.72)
p = 34 50 = . 68
o o Note: Different random numbers would have identified a different sample which would have resulted in different point estimates. Slide 13 Note: Usually Note: Usually we never know the population parameter so we cannot compute compute the sampling error. (If we knew the population parameter, we would not bother estimating it with a sample.) But we can make probability statements about the sampling error, based on sampling sampling distribution. Example: “The survey has a margin of error of plus or minus 3 percentage points” usually means that “There is a 95% probability that the sample proportion is within +/- 3% of the population proportion.” +/- Slide 14 Sampling Sampling Distributions
A sampling distribution is the probability distribution of a sample statistic. o Example: the sampling distribution of the sample mean. • There are many possible sample that could be drawn from a given population. • The sample mean would vary from sample to sample. • The sampling distribution is the frequency distribution of the sample means obtained from all possible samples from the population.
o o What is a Sampling Distribution of x ?
Example • a) estimating average height of UCF students • b) row #1/sample #1 mean = 5’4” • c) row #2/sample #2 mean = 5’5” 5’5” • d) row #3/ row #3/sample #3 mean = 6’9” (!) #3 6’9” • e) etc • f) DISTRIBUTION OF ESTIMATES (means) • (1) 4’4”, 5’5”, 5’7”, 5’8”, . . ., 6’3”, 6’5”, 6’9” • So each sample gives a different sample mean. • (Note these should be random samples to use sampling theory.)
Slide 16 Slide 15 15 Distribution Distribution of Average Heights
f(x) Sampling Distributions
The sampling distribution is not the distribution of heights, but the distribution of the mean heights obtained from many different samples. o If we follow good sampling protocol, then most samples would give us a sample mean that is close to the true but unknown population mean. o But some samples, despite being random, will not be representative. • Some will include too many tall people, and give us a sample mean much larger than the population mean. • Others will include too many short people, and give us a sample mean much smaller than the population mean.
o Each sample gives one value 4’4” μ x
6’9” Point: Taking many samples yields a distribution of x values
Slide 17 Slide 18 18 3 Distribution of Average Heights
f(x) Central Limit Theorem
The basic idea of the Central Limit Theorem is that, as the sample size increases, the sampling distribution of the sample mean • Approaches a normal distribution, • With a mean equal to the population mean, • And a standard deviation equal to the population standard standard deviation divided by the square root of sample size. o Even if the population is not normally distributed, for large sample size, the sampling distribution of the sample mean is approximately normal.
o Each sample gives one value 4’4” μ x
6’9” Point: This is the sampling distribution of x values
Slide 19 Slide 20 20 Distribution Distribution of Average Heights
f(x) Sampling Distribution of x
The sampling distribution of x • is the probability distribution of all possible values of x from all possible samples o By the CLT, E( x ) = μ where: μ = the population mean and
o Each sample gives one value 4’4” μ x
6’9” x= ∑x
n i Point: This is the sampling distribution of x values
Slide 21 Slide 22 Sampling Distribution of x
o Sampling Distribution of x
o Standard Deviation of x Finite Population Infinite Population Standard Deviation of x σx = σ n standard error of the mean σx = ( σ
n ) N −n N −1 σx = σ
n • NOTE: A finite population is treated as being infinite if n/N < .05. • ( N − n ) / ( N − 1) is the finite correction factor. • σ x is referred to as the standard error of the mean.
Slide 23 Suppose σ = 48 & n = 36. σ x = 8 Suppose σ = 48 & n = 64. σ x = 6 Suppose σ = 48 & n = 100. σ x = 4.8 48 100 What happens to σ x as sample size grows? Point: As sample size grows, standard error of mean shrinks. As a result, larger sample size provides higher probability that sample mean is within specified distance of pop. mean. Slide 24 4 Sampling Distribution of x
o Sampling Sampling Distribution of x
If we use a large (n > 30) simple random sample, • the CENTRAL LIMIT THEOREM enables us to conclude that the sampling distribution of x can be approximated by a normal probability distribution distribution. o When the simple random sample is small (n < 30), • the sampling distribution of x can be considered normal normal only if we assume the population has a normal normal probability distribution. o Whenever the population has a normal probability distribution, • the sampling distribution of x is a normal probability distribution for ANY sample size
Slide 26 Standard Deviation of x Suppose σ = 48 & n = 64. σ x = 6 95% of all values x between 188 and 212. 95% of all samples give x within 2 σ of mean (=200, say). Suppose σ = 48 & n = 36. σ x = 8 95% of all possible x between 184 and 216. . Suppose σ = 48 & n = 100. σ x = 4.8 95% of all values x between 190.4 and 209.6. . Point: As sample size grows, 95% of all values closer and closer to mean . . . or more values closer to mean.
Slide 25 Sampling Distribution of x
Now have completely described sampling _ distribution of x _ • Use estimator such that E( x ) = μ • Calculate standard error of the mean • Determine shape of sampling distribution • Likely normal l
o o Example: ESPN
Sampling Distribution of x for the SAT Scores NOTE THIS σx = σ
n = 80 = 11. 3 50 E(x ) = μ
NOTE: two possible sources for value of σ: 1. Know it from population (e.g., see slide #13) 2. Estimate it using sample standard deviation x Slide 27 Slide 28 Example: Example: ESPN
o Example: ESPN
o Sampling Distribution of x for the SAT Scores n = 50 Sampling distribution of x Sampling Distribution of x for the SAT Scores What is the probability that • a simple random sample of 50 applicants (n = 50) • will provide an estimate of the population mean SAT score (μ) • within (plus or minus) 10 of μ ? Note: Note: assume that don’t know value of μ. Area = ?? Area = ?? μ - 10 μ μ + 10 x Slide 29 Slide 30 5 Example: ESPN(cont.)
•NOTE: • Pop std. dev was calculated = 80 (previous slide) • Std. dev. of x was calculated = 11.3 (previous slide) • USE STD. DEV. OF x = 11.3 •Here are the steps to find the answer: •Step (1) convert the x value of (μ – 10) to a Z-value: • z1= (μ – 10) - μ / 11.3 = -10 / 11.3 = -0.88 • so, x = (μ – 10) equivalent to z=-0.88 z=•Step (2) convert the x value of (μ + 10) to a Z-value: Z• z2 = (μ + 10) - μ / 11.3 = 10 / 11.3 = 0.88 • so, x = (μ + 10) equivalent to z= 0.88
Slide 31 Example: ESPN(cont.) HINT: draw picture
•Step (3) P(-0.88 < Z < 0.88) is area between P(z = - 0.88 and z = 0.88 • this area is sum of two areas: • area between z = - 0.88 & z =0 (area A1) & (area • area between z = 0 & z = 0.88 (area A2) •Step (4) P(-0.88 < Z < 0.88) = area A1 + area A2 P(area = 0.3106 + 0.3106 = 0.6212 •This answer means there is 62.12 % probability that x will be between (μ – 10) & (μ + 10) (or 62.12 % of all x values fall between (μ – 10) & (μ + 10) - from n = 50 sample). sample).
Slide 32 Example: ESPN
o Sampling Distribution of x
What is the probability that a simple random sample of 50 applicants will provide an estimate of the population mean SAT score that is within (plus or minus) 10 of the actual population mean μ ? • 62.12 % probability that x falls between (μ – 10) & (μ + 10) σ 80 o What if n = 256? σ= = =5
x Sampling Distribution of x for the SAT Scores from n = 50 sample Sampling distribution of x Area = .3106 Area = .3106 n 256 μ - 10 μ μ + 10 x There is 62.12% probability that x will be between (μ – 10) & (μ + 10) with n = 50
Slide 33 • 95.45 % probability that x falls between (μ – 10) & (μ + 10) Slide 34 Sampling Sampling Distribution of p
The sampling distribution of p is the probability distribution of all possible values of the sample proportion p .
o o Sampling Distribution of p
Standard Deviation of p Finite Population Infinite Population We want E ( p ) = p where: p = the population proportion σp = p (1 − p ) N − n N −1 n σp = p (1 − p ) n • σ p is referred to as the standard error of the proportion proportion. Slide 35 Slide 36 6 When Use Finite Pop./Infinite Pop. Formula?
Formulas for σ
x Example: ESPN
o and σ Sampling Distribution of Inp for In-State Residents .72 = population proportion p Is n/N < .05 ? YES use INFINITE population formula NO use FINITE population formula
NOTE THIS σp = .72(1−.72) =.0635 50 E ( p ) = p = . 72
The normal probability distribution is an acceptable approximation since np = 50(.72) = 36 > 5 and n(1 - p) = np 50(.28) = 14 > 5.
Slide 37 Slide 38 Example: ESPN
o Example: ESPN
o Sampling Distribution of p for In-State Residents InWhat is the probability that a simple random sample of 50 applicants will provide an estimate of the population proportion of in-state residents that is inwithin plus or minus .05 (5 percentage points) of the (5 actual population proportion? In other words, what is the probability that p will be between .67 and .77? Sampling Distribution of p for In-State Residents InSampling distribution of p Area = .2852 Area = .2852 2852 0.67 0.72 0.77 p For z = .77-.72/.0635 = .79, the area = (.2852)(2) = .5704. .77The probability is .5704 that p will be within p +/+/-.05 of the actual population proportion.
Slide 39 Slide 40 Hypothesis Testing Hypothesis Testing o Generally, any formal or informal testing begins with an idea, theory, speculation, guess, hunch or hypothesis about the population of interest • what is not true • what is true In formal hypothesis testing, there will always be two hypotheses: • the null hypothesis (H0) and • the alternative hypothesis (HA.) Slide 41 41 Slide 42 42 7 Hypothesis Testing (cont.)
The null hypothesis • is often (but not always - see example later) … o the idea that you think is not true.
o o o Example Example (cont.)
In our system of justice, the presumption about the guilt or innocence of the accused is . . . In other words, the null hypothesis is that the accused is ______________ and it is up to the prosecution to disprove the null hypothesis. (So, it appears that the prosecution believes that the null hypothesis is wrong.) o Tip: • Begin stating your hypotheses by first … i fi • putting what you believe is true in the alternative what is hypothesis hypothesis o Slide 43 Slide 44 44 Hypothesis Hypothesis Testing (cont.)
After testing the null hypothesis, you must draw one of two conclusions about it. Either the evidence favors the idea that the null hypothesis is • false (so you reject it in favor of the alternative) or • it is true (so you do NOT reject it.) o Notice that the two conclusions always refer to the null null hypothesis.
o o Example
A jury or judge will conclude either that the accused or is ??? or is not ???. Slide 45 45 Slide 46 46 Example
A jury or judge will conclude either that the accused or is ??? or is not ???. (Ever wonder why the second verdict is never "innocent"?) o In terms of hypothesis testing, the jury either • rejects the null hypothesis of innocence (accused is guilty) or • doesn't reject it (accused is not guilty.)
o o •Hypothesis Testing (cont.)
Now, it is obvious that you may make either • correct decisions or • incorrect decisions about the null hypothesis. Slide 47 47 Slide 48 48 8 Hypothesis Testing (cont.)
Correct decisions are: (H0: accused innocent) (H innocent) (in formal terms) Example reject false null convict guilty person don't reject true null acquit innocent person Incorrect decisions are: (in formal terms) reject true null don't reject false null
o Hypothesis Hypothesis Testing (cont.)
It is important to emphasize that you will never be 100% certain that you have made the correct decision. Because you are using a sample, not the population, there will always be a chance of making a wrong decision. making By the way, what are the sample & population in a trial? sample: evidence presented population: the complete truth o Example convict innocent person acquit guilty person o o o Slide 49 Slide 50 50 Type I & Type II Errors
o o Hypothesis Testing (cont.)
Over the years, researchers have concluded that it’s easier to control the chance of making the first type of incorrect decision (Type I error: rejecting a true null, e.g. convicting an innocent person) o So, every test of a hypothesis is conducted so as to control the probability of making the first t pe bilit fi ty of of error ("Type I error") – typically by making the probability of Type I error small. o For a given probability of Type I error, we prefer a test with a lower probability of a Type II error (for a given significance, we prefer greater power).
Slide 52 52 the error of rejecting a true null hypothesis is called Type Type I error the error of not rejecting a false null hypothesis is Type called Type II error Slide 51 Hypothesis Hypothesis Testing (cont.)
When the jury or judge convicts a person, there could be a 1% chance that they are convicting an innocent person person. o That is, there is a 1% chance that they have rejected a true null hypothesis of that person's innocence. o That is, the CHANCE OF MAKING TYPE I ERROR is 1%.
o The probability of making a Type I Error is called the level level of significance of the test. Slide 53 53 Slide 54 54 9 Hypothesis Testing (cont.)
Three Key Definitions The level of significance is a probability that level • we set before doing testing • we want to be small: usually 10%, 5% or 1% • is max. probability of making Type I error that we will tolerate The software calculates a p-value • "p" stands for probability • This should be compared with the level of significance (above) The value of (1 - the p-value) is the level of plevel confidence you have in a test's conclusions.
Slide 55 Hypothesis Hypothesis Testing (cont.)
When you conduct a test using a PC, the statistical software will usually print a p-value for each test you pdo. o This value can range from 0 through 1.00 (0% to 100%.) o This tells you the likelihood that you will make a Type I Error if you reject the null (= the probability of rejecting the null, if it is in fact true).
o o o o Slide 56 56 Hypothesis Hypothesis Testing (cont.)
o Who Uses This Additionally, the corresponding level of confidence tells you how confident you are that your rejection of the null is correct. o Stock market analysts have shown that concentrated mutual funds (those with less than 25 stocks in their portfolio) had statistically significant better returns statistically nonduring 1998 than non-concentrated mutual funds. (Source: The Wall Street Journal, April 22, 1999,p.4.) The Slide 57 Slide 58 58 Who Uses This Who Uses This • Medical researchers have shown, using one-tailed onehypothesis tests, that the much-touted family of muchantidepressants including Prozac are slightly worse in most aspects than the older generation of antidepressants. (Source: The New York Times, March The 20, 1999.) • Researchers state that there is not statistically significant evidence that the much-touted vitamin muchginkgo biloba would improve memory. “The jury is The still out.” (Source: The New York Times, April 4, 1999 p.1.) Slide 59 59 Slide 60 60 10 Who Uses This Example • A study for Morgan Stanley Dean Witter examined stock fund performance over two consecutive fivefiveyear periods. The report found that of those in the top quartile in the first period, only...
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