Minitabl lab notes section 2

Statistics-10030 MINITAB – Lab2Small Sample Tests of Hypothesis About a Population Mean1.When dealing with large samples we can use the Central Limit Theorem to testhypotheses about a population mean. What do we do when we are dealing with small samples (i.e. n < 30), and therefore the normality of the sampling distribution of the mean does not follow from the Central Limit Theorem?In such cases we may still proceed providing that the underlying distribution from which the sample is drawn is normal .However, even when the population from which the sample is drawn may benormal, using the sample standard deviation, s, as an estimate for σ, the population standard deviation is problematic. To correct for this we conduct our hypothesis test as before except a distribution called the t distribution (you may see this called the student t distribution in certain texts) is used in place of the normal distribution for choosing a rejection region.The test statistic is:ns x t μ-=and a rejection region is chosen at the desired α level from the t distribution with n-1 degrees of freedom.How to interpret the Rejection Region:A major car manufacturer wants to test a new engine to determine whether it meetsair-pollution standards. The mean emission μ of all engines of this type must be less than 20 parts per million of carbon. Ten engines are tested to establish that the population of engines will be in line with the standards. The results are:15.6, 16.2, 22.5, 20.5, 16.4, 19.4, 16.6, 17.9, 12.7, 13.9 .The manufacturer is making the assumption that the relative frequency histogramof the emission levels for the population of engines of this type is approximately normal. Enter this data into MINITAB labelling the variable appropriately and answer the following question using α = .01.Do the data supply sufficient evidence to allow the manufacturer to conclude that this type of engine meets the pollution standards ?Here are the familiar steps in hypothesis testing - amended to reflect small samples.Step 1. Choose the population characteristic of interest . μ - the population mean Step 2. Choose the significance level . α = .01 (or 1% level). Step 3. State null hypothesis . Ho: μ = 20 ppm Step 4. State alternative hypothesis . Ha: μ < 20 ppmStep 5. Choose a test statistic . In this case the test statistic chosen isns x t μ-=Step 6. Choose a rejection regionSince the alternative hypothesis includes all means of less than 20 ppm, this is a one-tailed test. The rejection region will be in the lower tail of the t, df=9 distribution. First get the t quantile (critical value) such that 1% of the t distribution with n-1 degrees of freedom is to the right and therefore 99% is to the left with the following command and then take the negative of the answer since this is a lower tailed test,MTB > INVCDF .99; SUBC> t 9.W hat is the answer ? _____________So the null hypothesis will be rejected if:Step 7. Calculate the test statisticFill in the following equation.=-=-=-=ns x s x t xμμ ___________Step 8. State Conclusion in the context of the questionReject / Fail To Reject the Ho: at α = _______, that____________________________________________________________________________________________________2. MINITAB has a built in function for small sample hypothesis testing about a populationmean Repeat the above test using this function. Go to the STAT - BASIC STATISTICS -1 Sample t. Specify the Null hypothesis mean in the 'test mean' box and click options.Choose the correct confidence level and alternative hypothesis and click ok twice.MINITAB will now print the result of the test in the session window. You will see N - the number in the sample, Mean - the sample mean, Stdev - the standard deviation of the sample, SE Mean - the standard error of the mean, 100(1-α)% CI or upper or lower bound (i.e. a two sided or one sided confidence interval depending on the Ha:), t - the test statistic, P - the actual probability that the sample mean came from the population specified by the H o.What is the P value for this hypothesis test ? ____________.Reject / Fail To Reject the Ho: at α = _______, that____________________________________________________________________________________________________Change the Ha: to a 2-sided test and redo the test. What is the two sided 99%confidence interval for the mean ? __________________________________.Has the P value changed form last time, what is it now ? ________If so explain why.Assignment: Due 2 weeks, Monday 27th FebruaryQ1Open the worksheet called Lab 2, which is on the Minitab class page. Read the information below regarding each sample and then using this weeks and last week’s lab sheets, perform appropriate hypothesis tests for each one.[ Hint: Generate the summary statistics for each sample first to familiarise yourself withthe data.]Sample 1:The average cost of a loaf of bread is € 0.89. Loaves of bread were bought in ten shopsin Dublin to find evidence against this claim. Sample 1 contains these costs.[Perform this test at 99% confidence level]Answer:a. State the H o and H a hypothesis.b. State the α and number of subjects in this experimentc. Should a Z test or a t test be performed here?d. What is the (i) test Statistic, (ii) Critical value and (iii) p-value for this experiment.e. State your results and conclusions in relation to the question.Q2Sample 2:Students at a particular secondary school were complaining about the time it took themto travel to school on public transport and want the school to organise a private schoolbus. The board of management insisted that the journey on public transport only took anaverage of 20 minutes. The students conducted a survey of 50 students who travel bypublic transport in an effort to prove the journey takes longer. Sample 2 contains thetime in minutes it took each one to travel.[Perform this test at 95% confidence level]Answer:a. State the H o and H a hypothesis.b. State the α and number of subjects in this experiment.c. Should a Z test or a t test be performed here?d. What is the (i) test Statistic, (ii) Critical value and (iii) p-value for this experiment.e. State your results and conclusions in relation to the question.Q3Sample 3:An experiment was conducted to examine the weight of a batch of flower bulbs in agardening shop. If the average weight is less than 3 grams then the whole batch will bethrown out. A sample from the batch were taken at random and weighed and recorded in Sample 3.[Perform this test at 95% confidence level]Answer:a. State the H o and H a hypothesis.b. State the α and number of subjects in this experiment.c. Should a Z test or a t test be performed here?d. What is the (i) test Statistic, (ii) Critical value and (iii) p-value for this experiment.e. State your results and conclusions in relation to the question.Q4Answer these questions:a. When a set of data has a t-distribution with 12 df what is the area to the left of the point0.47?b. When a set of data has a t-distribution with 6 df what is the area to the right of the point1.25?c. When a set of data has a t-distribution with 6 df what is the area to the right of the point1.25?d. When a set of data has a t-distribution with 16 df what is the t-value with an area of 0.45to the left of this point?e. When a set of data has a t-distribution with 5 df what is the t-value with an area of 0.89to the left of this point?f. When a set of data has a t-distribution with 7df what is the t-value with an area of 0.79 tothe right of this point?REVISION SUMMARYAfter this lab you should be able to:-Understand the 8 steps of a hypothesis test-Look up t-critical values (with appropriate degrees of freedom) in theCambridge tables and Minitab-Perform a 1-sample t hypothesis test using Minitab’s inbuilt function-Interpret a hypothesis test from comparing the test statistic and the critical value-Interpret a hypothesis test from comparing the p-value and α level-Tell when you should perform a z-test and a t-testEND。