Statistical Data Processing – TEST – Part 1 – 30 points Name: ………………………………… 1) Descriptive statistics: Count the measures of central tendency and measures of variability. 25 25 22 28 31 23 26 26 23 29 29 25 Population mean: Mode: Median: Range: Population variance: Population standard deviation: 2) A company wants to know if customer preferences for five types of products—Product A, Product B, Product C, Product D, Product E—match the market's expected preference distribution. They believe that 30% of customers prefer Product A, 20% prefer Product B, 20% prefer Product C, 20% prefer Product D, and 10% prefer Product E. They conduct a survey of 500 customers to test this hypothesis. Test at the alpha significance level 0.1 Null hypothesis (H0​): Customer preferences match the expected distribution. Alternative hypothesis (H1): Customer preferences do not match the expected distribution Oi (Observed) Ei (Expected) Test criterion Product A 110 Product B 100 Product C 90 Product D 110 Product E 90 Test criterion = Critical Value = Do you reject or fail to reject the null hypothesis? Why? 3) A researcher wants to determine if there is an association between gender and preference for two types of books. The researcher surveys 150 people, and the data collected is summarized in the table below Book 1 Book 2 Total Male 40 35 Female 30 45 Total Test at the alpha significance level 0.01 Null hypothesis (H₀): Gender and book preference are independent (no association). Alternative hypothesis (H₁): Gender and book preference are not independent (there is an association). Expected frequencies Book 1 Book 2 Male Female Test criterion Book 1 Book 2 Male Female Test criterion = Critical Value = Do you reject or fail to reject the null hypothesis? Why? 4) Analysis of variance: 1st car 2nd car 3rd car 4.851 5.899 3.952 4.745 4.256 4.052 4.781 4.052 3.825 4.782 4.125 3.932 Textové pole: We tested the mileage of three cars, and you are given three samples of observations. • At the significance level of 𝛼 = 1 %, test the null hypothesis that all three cars have the same results. • Formulate the null hypothesis 𝜇₁=𝜇₂=𝜇₃ that all three cars have the same mileages. • Formulate the alternative hypothesis that the mileage of at least one mileages is different. • Calculate: — the group sums and the group sample means 𝑦¯𝑖 — the grand sum and the grand sample mean 𝑦¯ • Calculate the quantity SSB = the sum of squares “between” = RegSS. • Calculate the quantity SSW = the sum of squares “within” = RSS. • Calculate the Coefficient of Determination (𝑅² = RegSS / TSS, where TSS = RegSS+RSS). Is the fit “good”/“poor”? • Determine the quantity DFB = the degrees of freedom “between”. • Determine the quantity DFW = the degrees of freedom “within”. • Calculate the quantity MSB = the mean squares “between”, i.e. the the sample variance between the groups = SSB / DFB. • Calculate the quantity MSW = the mean squares “within”, i.e. the the sample variance within the groups = SSW / DFW. • Calculate the statistic 𝐹 = MSB / MSW = (SSB / SSW) / (DFB / DFW). • Calculate the critical value 𝑐. • Calculate the p-value of the test. • Do you reject or fail to reject the null hypothesis?