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Introduction to Statistical Methods

by: Ena Kris

Introduction to Statistical Methods STAT 301

Marketplace > Colorado State University > Statistics > STAT 301 > Introduction to Statistical Methods
Ena Kris
GPA 3.91

Brett Hunter

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Brett Hunter
Class Notes
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This 8 page Class Notes was uploaded by Ena Kris on Tuesday September 22, 2015. The Class Notes belongs to STAT 301 at Colorado State University taught by Brett Hunter in Fall. Since its upload, it has received 19 views. For similar materials see /class/210329/stat-301-colorado-state-university in Statistics at Colorado State University.


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Date Created: 09/22/15
26 September Sampling Distribution of the Sample Mean The sampling distribution of the sample mean is a probability distribution consisting of all the sample means of a given sample size selected from a population Consider two scenarios You know the population distribution is normal Sampling distribution of is normal We don t know the distribution of the population or we know it s not normal Sampling distribution of is normal if n is quotlargequot Properties of the Sampling Distribution of is the mean of the observations in a random sample of size n from a population with mean u and standard deviation 0 The mean of the distribution of is u and the standard distribution is 039 M l1 7 52 039 79039 V W When n is significantly large the sampling distribution of is approximately normal by Central Limit Theorem Large enough if n 2 30 Central Limit Theorem Regardless of the shape of the population distribution the shape of the sampling distribution of the sample mean approaches a normal distribution as n increases As n increases the normal approximation of the sample distribution of gets better Probabilities for T Since Nu 0 we can standardize our values using And then use standard normal tables to find probabilities Standard error ofE S V W S is sample standard deviation Confidence Intervals for u Two scenarios 0 is known Rarely plausible o is unknown Constructing a CI for u when 0 is known Recall general formula point estimate i critical value x standard error Standard deviation known when 0 known use instead of standard error Point estimate E Critical value 2cm since E quot Nu 0 I Standard deviation 039 Formula E i 20 Margin of error Bound on error Constructing a CI for u when 0 is unknown Point estimate 2 h 2 02f gt x Nu 0 w ere IE 7 or n 30 5 Since 0 Is unknown we estImate the standard deVIatIon using 5 E standard error Previously critical values based on fact that z Y N0 1 7 5 If we standardize here we get Which does not have a standard normal distribution Has a tdistribution 23 September Determining Sample Size Let B be the bound on the error of estimation another name for the margin oferror So b critical value x standard error Often a researcher wants to achieve a certain bound on the error How do we do this Determine the desired confidence level Estimate population variance square of SE Determine required sample size 3 factors that determine n Confidence level Maximum allowable error B Variation in population We find the minimum sample size required for each of the desired values that determine n Determining sample size for a proportion n p LNG n minimum sample size p estimation of population proportion 2 critical value corresponding to confidence level B maximum allowable error We know that B 2 lg Most times won t have a value for p In such cases we let p5 which gives the most conservative value of n Son 25 B 4Bz Note When performing sample size calculations ALWAYS round the sample size up to the next whole number 12 October Comparing Two Populations or Treatments Checking for differences in population means 2 situations Paired data Two samples are paired if observations from one sample are paired in some meaningful way with observations from the second sample Two independent samples The selection of objects that make up one sample does not influence the selection of objects in the other sample Paired Data Set Up eg A researched wishes to determine whether aerobic exercise can decrease blood pressure so a measure is taken on each patient before and after participating in an aerobic exercise program We can pair the before and after measurements by patient The advantage of paired data is to eliminate extraneous effects caused by variability with the objects ie patients themselves Look at the lldifference populationquot the difference between an observation in population 1 and its paired observation in population 2 Let ud mean value of the difference population Let 0 standard deviation of the difference population Let Ydor E sample mean of difference population Let sd sample standard deviation of difference population Once we have a difference population or difference sample worked out we can perform a one sample hypothesis test and compute one sample Cs on the difference sample Hypotheses Upper Tail H0 udS hypothesized value ud0 Pd 1 Lower Tail H0 udz hypothesized value ud0 HA udlt TwoTail H0 ud hypothesized value ud0 HA udat Test Statistic Case 1 odknown n 2 30 or both populations normal Yd do N mo 1 7d 21951 n Case 2 od unknown n 2 30 or both populations normal Yd Mn N Li tlesl tnrl 5 Note 1 If both populations are normal the population of differences is normal Note 2 pvalues found in same manner as one population hypothesis test for u Confidence Intervals Case 1 CI d i Zcril 5d tcril n CI Yd i 2 Case 9 November Regression Assume we have bivariate data with variables X and Y where X and Y are both numeric Question of interest Is there a relationship between X and Y If there is How strong is it What type of relationship is it Can I predict Y using X First step is exploring a possible relationship Construct a scatterplot Is the relationship positive or negative Positive As X increases Y also increases Negative As X increases Y decreases


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