### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# 260 Class Note for STAT 30100 at Purdue

### View Full Document

## 19

## 0

## Popular in Course

## Popular in Department

This 12 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Purdue University taught by a professor in Fall. Since its upload, it has received 19 views.

## Similar to Course at Purdue

## Reviews for 260 Class Note for STAT 30100 at Purdue

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 02/06/15

Chapter 6 Con dence Intervals and Hypothesis Testing Using Z for the CI and test of the sample mean Why do we even bother analyzing data We want to draw conclusions from the data Why can t we just accept our sample mean or sample proportion as the official mean or proportion for the population Every time we estimate the statistics f p sample mean and sample proportion we get a different answer due to sampling variability Two most common types of formal statistical inference 0 Con dence Intervals when we want to estimate a population parameter 0 Signi cance Tests when we want to assess the evidence provided by the data in favor of some claim about the population yesno question about the population Con dence Intervals allow us to estimate a range of values for the population mean or population proportion The true mean or proportion for the population exists and is a xed number but we just don t know what it is Using our sample statistic we can create a net to give us an estimate of where to expect the population parameter to be Con dence interval net Population parameter invisible stationary butter yamp We don t know exactly where the butter y is but from our sample we have a pretty good estimate of the location Density curve of If we just take a single sample our single con dence interval net may or 39 may not include the population parameter However if we take many samples of the same size and create a con dence interval from each sample statistic over the long run 95 of our confidence intervals will contain the true population parameter if we are using a 95 confidence level Figule s a lnlludurh un m wrung1 mu mude l K SWHIVMMqum r If you increase the sample size n you decrease the size of your net or your margin of error n320 n1280 14000 16000 18000 20000 22000 24000 26000 If you increase your con dence level C then you increase the size of your net or your margin of error 99 confidence 95 confidence 14000 16000 13000 20000 22000 24000 26000 nmu mmquotmmmm hmnmmquot monv Klmmm wimnvww A smaller net is good because it gives you more information It is a smaller range for Where to expect your true population parameter Freeman applet G0 to emrse website Freeman link statistical applets con dence interval Con dence intervals look like estimate 1 margin of error Con dence Interval for a Population Mean 11 z n Where 2 is the value on the standard normal curve with area C between 72 and 2 Table D at the back of the book also contains more 2 values on the bottom row 2 1645 1960 2576 C 90 95 99 Remember from Ch 5 that the mean and standard deviation for a sample mean are 2le 5 x 5 Also remember that if X is normally distributed then X will be too and if n is large the sample mean will be approximately normally distributed even if X is not normally distributed Central Limit Theorem What if your margin of error is too large Here are ways to reduce it 0 Increase the sample size bigger n 0 Use a lower level of con dence smaller C 0 Reduce 0 Sample Size n for Desired Margin of Error m 2 n 2 ax m Note that it is the sample size n that in uences the margin of error The population size has nothing to do with it O Be careful You can only use the formula x i Z quot under certa1n circumstances n Data must be an SRS from the population Do not use if the sampling is anything more complicated than an SRS Data must be collected correctly no bias The margin of error covers only random sampling errors Undercoverage and nonresponse are not covered Outliers can have a big effect on the confidence interval This makes sense because we use the mean and standard deViation to get a CI You must know the standard deViation of the population at Examples 1 A questionnaire of drinking habits was given to a random sample of fraternity members and each student was asked to report the of beers he had drunk in the past month The sample of 30 students resulted in an average of 22 beers with a population standard deviation of 9 beers a Give a 90 con dence interval for the mean number of beers drunk by fraternity members in the past month b Is it true that 90 of the fraternity members each month drink the number of beers that lie in the interval you found in part a Explain your answer c What is the margin of error for the 90 confidence interval d How many students should you sample if you want a margin of error of l for a 90 confidence interval 2 A sample of 12 STAT 301 students yields the following Exam 1 scores 78 62 99 85 94 53 88 90 86 92 75 92 Assume that the population standard deviation is 10 The sample mean can be calculated using SPSS or calculator to be 8283 Note Do NOT use any SPSS con dence intervalsithey are good only for Chapter 7 not this type of CI You must get these Z confidence intervals by hand a Find the 90 con dence interval for the mean score u for STAT 301 students b Find the 95 con dence interval c Find the 99 con dence interval d How do the margins of error in b c and d change as the con dence level increases Why Hypothesis Testing To do a signi cance test you need 2 hypotheses o Hg Null Hypothesis the statement being tested usually phrased as no effec or no difference 0 Ha Alternative Hypothesis the statement we hope or suspect is true instead of Hg Hypotheses always refer to some population or model Not to a particular outcome Hypotheses can be onesided or twosided o One sided hypothesis covers just part of the range for your parameter Hg u 10 OR Hg u 10 Ha ugt 10 Ha lt 10 o Two sided hypothesis covers the whole possible range for your parameter Hg u 10 Ha ui 10 Even though Ha is what we hope or believe to be true our test gives evidence for or against Hg only We never prove Hg true we can only state whether we have enough evidence to reject Hg which is evidence in favor of Hg but not proof that Ha is true or that we don t have enough evidence to reject Hg Example Exercise 63 7 p 418 Each of the following situations requires a significance test about a population mean u State the appropriate null hypothesis H0 and alternative hypothesis Ha in each case a Census Bureau data show that the mean household income in the area served by a shopping mall is 72 500 per year A market research rm questions shoppers at the mall to nd out whether the mean household income of mall shoppers is higher than that of the general population b Last year your company s service technicians took an average of 18 hours to respond to trouble calls from business customers who had purchased service contracts Do this year s data show a different average response time A test statistic measures compatibility between the H g and the data P Value the probability computed assuming that H0 is true that the test statistic would take a value as extreme or more extreme than that actually observed due to random uctuation It is a measure of how unusual your sample results are 0 The smaller the Pvalue the stronger the evidence against H 0 provided by the data 0 Calculate the Pvalue by using the sampling distribution of the test statistic only the normal distribution for Chapter 6 0 Compare Pvalue to a significance level a o If the P Value S a we can reject Ho 0 If you can reject Ho your results are signi cant 0 If you do not reject Ho your results are not signi cant T1124 Slsz cnmmnn to alltesls or signi cance 1 2 State the null hypothesrs Hg and the altematrye hypothesrs Ha Caleulate the value ofthe test staustre zrscore m Chapter 6 Draw a preture ofwhat Ha looks lrke and nd the Prvalue use the normal table m Chapter 6 State your eonelusron about the datarn a sentenee uslng the Prvalue andor eompanng the Prvalue to a slgmflcance level for your eyrdenee z enfnrzl ngul nMean To te unkn stthe hypothesrs Hn a tlnbased on an SRS ofslze n from apopulauon wth own mean and known standard deyrauon a n a in the Prvalues for atest oley agarnst eompute the test stausue zn pdnrsazzzd H ltglsPZSZg glsZPZB Zg These Prvalues are enaet lfthe populatron ls normally dstnbuted and are apprommately eorreet for large n m other eases Examples 1 Last year the government made a claim that the average income of the American people was 33950 However a sample of 50 people taken recently showed an average income of 34076 with a population standard deviation of 324 Is the govemment s estimate too low Conduct a signi cance test to see if the true mean is more than the reported average Use an 0c001 2 An agroeconomist examines the cellulose content of a variety of alfalfa hay Suppose that the cellulose content in the population has a standard deviation of 8 mg A sample of 15 cuttings has a mean cellulose content of 145 mg a A previous study claimed that the mean cellulose content was 140 mg Perform a hypothesis test to determine if the mean cellulose content is different from 140 mg if 0c005 b Find a 95 confidence interval for the mean cellulose content c Now try the test from part a again using the confidence interval from part b to do the hypothesis test The result should be the same How does a relate to con dence intervals If you have a 2sided test and if the a and confidence level add to 100 you can reject H0 if 0 the number you were checking is not in the confidence interval 10 3 An environmentalist collects a liter of water from 45 different locations along the banks of a stream He measures the amount of dissolved oxygen in each specimen The mean oxygen level is 462 mg with the population standard deviation of 092 A water purifying company claims that the mean level of oxygen in the water is 5 mg Conduct a hypothesis test with 0c0001 to determine whether the mean oxygen level is less than 5 mg Annual Drinking Water Quality Report 2004 Town of Brookston IN I m pleased to report that our drinking water is safe and meets federal and state requirements Test Results MCL is the maximum contaminant level the highest level of a contaminant that is allowed in drinking water Contaminant Violation YN Level Detected Unit MCL measurement Betaphoton N 21 i 3 2 mremyr 4 emitters Alpha emitters N 0 i 16 pCil 15 Barium N 0216 ppm 2 Copper N 0039 to 0453 ppm 13 Fluoride N 001 ppm 4 Sodium N 00 ppm NA One of these violation reports should actually be a yes instead of a no Which one is it and why What hypotheses go along with these confidence intervals Note When I called the town of Brookston of ce to ask them about this the water manager called the state EPA of ce to get more information What they told him was that yes technically I was correct but that they don t use the confidence intervals that 11 are reported Apparently these are the FEDERAL EPA rules They only use the mean I tried to get sample size or other information but wasn t able to learn anything more Pvalues can be more informative than a rejectdo not reject H 0 based on a As P value gets smaller the evidence for rejecting Ho gets stronger Just because we use 0 005 a lot doesn t mean that s the level you have to useiit s just the most common There s nothing particularly special about that level In a large sample even tiny deviations from the null hypothesis can be important If we fail to reject Ho it may be because H0 is true or because our sample size is insuf cient to detect the alternative Plot your data and look at Pvalue both to determine your conclusions Could outliers be part of the problem A con dence interval actually estimates the size of an effect rather than simply asking if it is too large to reasonably occur by chance alone You must have a welldesigned experiment in order for statistical inference to work Randomization is important 12

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "I used the money I made selling my notes & study guides to pay for spring break in Olympia, Washington...which was Sweet!"

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.