3 Topic 19 - Confidence Interval: Mean, σ is unknown The purpose of confidence intervals is to use the sample statistic to construct an interval of values that you can be reasonably confident contains the actual, though unknown, parameter.The estimated standard deviation of the sample statistic X-bar is called the standard errorConfidence Interval for a population proportion :where n >= 30t * is calculated based on level of confidenceWhen running for example 95% Confidence Interval:95% is called Confidence Level andwe are allowing possible 5% for error, we call this alpha (α )= 5% where α is the significant level
4 Topic 19 - Confidence Interval: Mean, σ is unknown Use if the sample data is given, use the Stat, Edit and enter data in the calculator before running the Confidence IntervalL1 is where data is entered by youC-Level: is the level you are running the Confidence IntervalUse if the information about sample data is given.X-Bar mean of sample dataSx is Standard deviation of the samplen is sample sizeC-Level: is the level you are running the Confidence Interval
6 Travel time to work. A study of commuting times reports the travel times to work of a random sample of 20 employed adults in New York State. The mean is = minutes and the standard deviation is s = minutes. What is the standard error of the mean?s/√n = 21.88/√20 = minutes.
7 Ancient air. The composition of the earth’s atmosphere may have changed overtime. To try to discover the nature of the atmosphere long ago, we can examine the gas in bubbles inside ancient amber. Amber is tree resin that has hardened and been trapped in rocks. The gas in bubbles within amber should be a sample of the atmosphere at the time the amber was formed. Measurements on specimens of amber from the late Cretaceous era (75 to 95 million years ago) give these percents of nitrogen:63.465.064.463.354.864.560.849.151.0Assume (this is not yet agreed on by experts) that these observations are an SRS from the late Cretaceous atmosphere. Use a 90% confidence interval to estimate the mean percent of nitrogen in ancient air.
8 Ancient air. Enter data for L1. 95% confidence Interval: Using TI83, under Stat, TEST,Choose option 8:TIntervalMean of the sample = 59.6Standard deviation = 6.26Degree of freedom = df= 8Confidence interval for mean percent of nitrogen is between 54.8 and 64.4.
11 Topic 20 – Test of Significant: Mean The purpose of Test of Significant is when we do know the population Parameter but we do not necessary agree with it or we have question about it. To do the test we need to run a sample and we use the statistic to test its validity.Step 1: Identify and define the parameter.Step 2: we initiate hypothesis regarding the question – we can not run test of significant without establishing the hypothesisStep 3: Decide what test we have to run, in case of proportion, we use t-test
12 Topic 20 – Test of Significant: Mean Step 4: Run the test from calculatorStep 5: From the calculator write down the p-value T-testStep 6: Compare your p-value with α – alpha – Significant LevelIf p-value is smaller than αwe “reject” the null hypothesis, then it is statistically significant based on data.If p-value is greater than the αwe “Fail to reject” the null hypothesis, then it is not statistically significant based on data.Last step: we write conclusion based on step 6 at significant level αp- value > 0.1: little or no evidence against H0• < p- value <= 0.10: some evidence against H0• < p- value <= 0.05: moderate evidence against H0• < p- value <= 0.01: strong evidence against H0• p- value <= 0.001: very strong evidence against H0
13 Few Possible cases to look at: A teacher suspects that the mean for older students is higher than 115Higher than means (> 115)The opposite of higher than is less than or equal to ( 115)Comparing the two, null hypothesis is the comparison that includes equality (=)Ho: µ = 115Ha: µ > 115One-sided alternativeA teacher suspects that the mean for older students is same or more than 115Same or more than means (> 115)The opposite of same or more than is less than (< 115)Ho: µ = 115Ha: µ < 115One-sided alternativeA teacher suspects that the mean for older students is also 115Same means (= 115)The opposite of same is not equal to ( 115)Ho: µ = 115Ha: µ 115Two-sided alternative
14 Fuel economy. According to the Environmental Protection Agency (EPA), the Honda Civic hybrid car gets 51 miles per gallon (mpg) on the highway. The EPA ratings often overstate true fuel economy. Larry keeps careful records of the gas mileage of his new Civic hybrid for 3000 miles of highway driving. His result is x-bar= 47.2 mpg. Larry wonders whether the data show that his true long-term average highway mileage is less than 51 mpg. What are his null and alternative hypotheses?AnswerLarry wonders whether the data show that his true long-term average highway mileage is less than 51 mpg.H0: µ = 51 mpg;Ha: µ < 51 mpg.
15 ProblemIf a researcher is interested in testing whether the mean is different from some claimed value, 55, then the null and alternative aretest the hypothesesH0: μ = 55,Ha: μ ≠ 55
16 H0 : x-bar = 1000 grams Ha : x-bar < 1000 grams Stating hypotheses. In planning a study of the birth weights of babies whose mothers did not see a doctor before delivery, a researcher states the hypotheses asH0 : x-bar = 1000 grams Ha : x-bar < 1000 gramsWhat’s wrong with this?Hypotheses should be stated in terms of µ, not x-bar .
17 Topic 20 – Test of Significant: Mean, σ is unknown Use if the sample data is given, use the Stat, Edit and enter data in the calculator before running the T-testµ0 is mean–value in questionList: L1 where the raw data is entered by youµ: is the alternative hypothesisUse if the information about sample data is given.µ0 is mean–value in questionX-bar is sample meanSx is Sample Standard deviationn is sample sizeµ: is the alternative hypothesis
18 Improving your SAT score. We suspect that on the average students will score higher on their second attempt at the SAT mathematics exam than on their first attempt. Suppose we know that the changes in score (second try minus first try) follow a Normal distribution. Here are the results for 46 randomly chosen high school students:Do these data give good evidence that the mean change in the population is greater than zero?−30244770−6255−41−32128−11−43122−105632−28−1911757−14−587727−3351−672994212−53−49498−2496120−2−3999
19 Activity 20- 2: Sleeping Times The null hypothesis is that the mean sleep time of the population is 7 hours. In symbols, the null hypothesis is H0 : µ = 7.0 hours. The alternative hypothesis is that the mean sleep time of the population is not 7 hours. In symbols, the alternative hypothesis is Ha : µ ≠ 7.0 hours.Sample NumberSample SizeSample MeanSampleSDTest Statisticp- value1106.60.82521.5973304
22 ProblemAssume that you are conducting a test of significance using a significance level of α = If your test yields a P-value of 0.08, what is the appropriate conclusion?P-value = 0.08 < 0.10 Reject Null, It is statistically significant
23 ProblemThe nicotine content in cigarettes of a certain brand is normally distributed with mean (in milligrams) μ and standard deviation σ = 0.1. The brand advertises that the mean nicotine content of their cigarettes is 1.5, but measurements on a random sample of 400 cigarettes of this brand gave a mean of x = Is this evidence that the mean nicotine content is actually higher than advertised?at significance level α = You concludeIs this evidence that the mean nicotine content is actually higher than advertised? State the hypothesistest the hypothesesH0: μ = 1.5,Ha: μ > 1.5
24 ProblemA researcher wants to know if the average time in jail for robbery has increased from what it was several years ago when the average sentence was 7 years. He obtains data on 400 more recent robberies and finds an average time served of 7.5 years. If we assume the standard deviation of sample is 3 years, what is the p-value of the test? at significance level α = You conclude