Two Population Means with Unknown Standard Deviations

10.1IntroductionIndepent1)Mean entry-level salaries for college graduates with mechanical engineering degreesand electrical engineering degrees are believed to be approximately the same, Arecruiting office thinks that the mean mechanical engineering salary is actually lowerthan the mean electrical engineering salary.The recruiting office randomly surveys 50 entry level mechanical engineers. and 60 entrylevel electrical engineers. The sample of mechanical engineers had a mean salary of$46,100 and a standard deviation of $3,450. The sample of electrical engineers had amean salary of $46,700 and a standard deviation of $4,210.Are the salaries of mechanical engineers Hower; on average, than mechanical engineers?2)A pharmaceutical company is interested in producing echinacea supplements. However,before going into production, they want to test the effectiveness of echinacea in reducingrhinovirus infections (viruses that typically cause common colds).In a test of the effectiveness of echinacea, 40 of the 50 subjects treated with echinaceadeveloped rhinovirus infections. In a placebo group, 88 out of the 103 subjects developedrhinovirus infections.How could the pharmaceutical company determine if their supplement is effective inreducing rhinovirus infections?3)A new fuel treatment was designed to increase fuel efficiency in vehicles. The fuelefficiency of the vehicles was measured before and after the fuel treatment was applied.Six vehicles were randomly selected and the data are shown in the table below. Thedifferences have a normal distribution.VehicleABCDEBefore22.531.225.119.827.6After24.135.827.220.135.4Do the data suggest that the fuel treatment is effective in increasing a vehicle's fuelefficiency? Section 10.1: Hypothesis Testing with Two Samples: Two Population Means with UnknownStandard DeviationsGoals:1) Determine if the samples are independent2) Know the requirements of working with two Independent samples3) Determine if the population variances can be assumed equal (the answer is Never assume thatalmost always "no")always no there equal4) Conduct a full hypothesis test putting together concepts from the previoussections.5) Use confidence intervals to fest a claim about the two population meansRequirements:1) The two independent samples are simple random samples from two distinctpopulations.2) Population standard deviations are not known:3)For the two distinct populations:a) If the sample sizes are small, the distributions should be normalb) If the sample sizes are large (greater than 30), the distributions do not need to benormal Steps of Hypothesis Testing:1) Determine if you are working with:meanssingle sample (Use T-Test or TInterval)>two independent samples (Use 2-Samp Test or 2-SampTint: booled = "no")>matched or paired samples (Use T-Test or TInterval)proportions>single sample (Use 1-PropZTest or 1-PropZInt)two independent samples (Use 2-PropZTest or 2-PropZInt)2) Identify the claim and write it as a mathematical statement.3) Use the claim to construct the hypotheses4) Understand the outcomes of a test and potential errors.5) Determine, based on the claim and sample data, which distribution (normal orStudent t)6) Analyze the sample data to generate the information you need to make a decisionabout the null hypothesis.7) Write a meaningful conclusion.Decision about the null hypothesis:Hypothesis Test:Reject H if p-value < a. (a.5) M RevenPopulation 1Use sample Steds G) toSampletest a claim about m7 3AChapter 10.1xPopulation 1MIR Pop meanPopulation 2unknownSampelSample 2Use theseXI S,two sampleX2 S2(xy X2) toIBtest aclaimaboutMr3M2 Testing a claim about Two Means (independent samples, sigma unknown)Working with two samples (each sample represents a distinct population)Population 1MPopulation 2 MrSampleSample2XI S,NFX2 S2N2Hypotheses:Null will always beAssume me - M2 > Mr - M2-0A OR A1Ho : m.=m2R RHo:m-M2=0HATwo tailedHa m, -M2 toHo:m = Mzor Ho:M, - -M2=0CHa : M,LM2Ha : M1-M2CONegative3Haim. = M2or Ha Mr-M2=OHand M2Ha M1-M270 Examples:Given two independent samples below, conduct a hypothesis test. Use a 0.05 significancelevel to test the claim that the means are different.e=0.05MI EM2Sample 1Sample 2X1 = 123.4X2 = 117.5S1 = 8.1S2=15.3n1 = 35n2 = 462 1 SampTTestDistribution: Eto cal theseOriginalclaim: MI , #M2Ho: M1-M2Statistic: Test t=2.24H1: M1 M M2P-Value: 0.02850.05Decision about H0:Reject HoConclusion in non-technical terms:Their is sufficent evidence to Support the claimthat the means or different