THEORETICAL PROBLEMS FOR THE FINAL EXAM
• (Empirical) Probability distributions, parameters
• Theoretical random variable distributions (discrete & continuous)
• Sampling distributions & CLT & LLN (large vs small sample?)
• Point and interval estimation + accuracy + sample size
Hypothesis testing
• One-sample t-test & Z-test for 1 mean
• One-sample chi-square test for 1 variance and SD
• One-sample binomial & Z-test for 1 proportion
• Two-sample t-test: for paired and independent samples (including normality tests, homogeneity of variance)
• Two-sample nonparametric tests for 2 distributions (Wilcoxon tests)
• Two-sample test for 2 proportions (Z-test and chi-square test)
• ANOVA vs. nonparametric ANOVA (Kruskal-Wallis test), post-hoc tests
• Chi-square contingency test for many proportions
• The null hypothesis is:
• Another name for the alternative hypothesis
• True with 95% probability
• Usually a statement you try to find evidence against
• Statistically significant
• If the p-value of a significance test is 0.02,
• Conclude that the null hypothesis and alternative are the same
• We reject the null hypothesis at the 5% significance level
• We fail to reject the null hypothesis at the 5% significance level
• Should use a different null hypothesis
• Which of the properties below is a useful property of the normal distribution?
• The area under curve is equal to 1
• It is symmetric about the mean
• It can be fully described by using the mean and sd
• All of the above properties are properties of the normal distribution
• In order to study whether music taste and IQ level are related, a random sample of 540 students were classified according to the type of music they prefer (Classical/Rock/Jazz) and their IQ level (Low/Medium/High).
A meaningful display of the data from this study will be:
• Side-by-side box plots
• A pie chart
• A histogram
• A scatterplot
• A two-way table
• The statistical procedure that should be used to analyze this data (from question no. 4.) is:
• A 2-sample t-test
• A paired t-test
• ANOVA
• A chi-square test
• Simple linear regression
• A company claims that its medicine, Brand A, provides faster relief from pain than another company’s medicine, Brand B. A researcher tested both brands of medicine on two groups of randomly selected patients; one group used Brand A, and the other group used Brand B. The researcher recorded the relief time (in minutes) for each patient. Test your research hypothesis at the 5% level of significance (alfa).
Brand A: 19; 25; 33; 40; 50; 29
Brand B: 10; 15; 40; 50; 30; 15; 20
• Null hypothesis: there are no differences between the time of relief between brand A and brand B
• Alternative hypothesis: there is a significant difference between the time of relief
• Samples are: PAIRED/INDEPENDENT
• Normality assumption is violated TRUE/FALSE
• Homogeneity of variances: YES/NO
• The statistical procedure that should be used to test this hypothesis is:
• A 2-sample t-test
• A paired t-test
• A Wilcoxon rank sum test
• A Wilcoxon signed rank test
• 1-way ANOVA
• 2-way ANOVA
• Kruskal-Wallis test
• A chi-square test
• Test statistic = 0.94226
• p-value= 0.3663
• Final conclusion: There are no significant differences between the time of relief between brand A and brand B.
• The department of motor vehicles wants to check whether or not drivers are impaired after drinking two beers. The reaction times in an obstacle course are measured for a group of 8 randomly selected drivers before and then after the consumption of two beers. Test your research hypothesis at the 5% level of significance (alpha).
BEFORE: 7; 20; 18; 9; 8; 11; 13; 15
AFTER: 18; 25; 22; 15; 14; 15; 19; 29
• Null hypothesis: There are no significant differences between reaction time before and after consumption of two beers
• Alternative hypothesis: There is a significant difference between reaction time before and after consumption of two beers
• Normality assumption is violated TRUE/FALSE
• The statistical procedure that should be used to test this hypothesis is:
• A 2-sample t-test
• A paired t-test
• A Wilcoxon rank sum test
• A Wilcoxon signed rank test
• 1-way ANOVA
• 2-way ANOVA
• Kruskal-Wallis test
• A chi-square test
• Test statistic = 5.5217____________
• P-value= 0.000886___________________
• Final conclusion: Reject null hypothesis. There is a significant difference between reaction time before and after consumption of two beers
• A meaningful display of the data from the study presented in the question no. 7. will be:
• Side-by-side box plots
• A pie charts
• A histogram
• A scatterplot
• A two-way table
• The effective life (in hours) of batteries is compared by material type (1, 2 or 3) and operating temperature: Low (-10 ̊C), Medium (20 ̊C) or High (45 ̊C). Twelve batteries are randomly selected from each material type and are then randomly allocated to each temperature level. The resulting life of all 36 batteries is presented below:
Temperature
Low (-10 C)
Medium (20 C)
High (45 C)
Material type
1
130, 155, 74, 180
34, 40, 80, 75
20, 70, 82, 58
2
150, 188, 159, 126
136, 122, 106, 115
25, 70, 58, 45
3
138, 110, 168, 160
174, 120, 150, 139
96, 104, 82, 60
Is there any significant difference in mean life of the batteries for various material type and operating temperature levels? Test your research hypothesis at the 5% level of significance (alpha).
• Null hypothesis: distribution of battery life is the same for all 3 temperatures and 3 material types
• Alternative hypothesis: there are significant differences in distribution of battery life for different materials and temperatures
• Normality assumption is violated TRUE/FALSE
• The statistical procedure that should be used to test this claim is:
• A 2-sample t-test
• A paired t-test
• A Wilcoxon rank sum test
• A Wilcoxon signed rank test
• 1-way ANOVA
• 2-way ANOVA
• Kruskal-Wallis test
• A chi-square test
• Test statistic(s) = Fmat=5.947 Ftemp=21.776
• p-value(s)= p1= 0.00651** p2= 0***
• Final conclusion: there are significant differences in distribution of battery life for different materials and temperatures
• It is known that about 20% of all income tax returns have errors. Government officials would like to know that percentage more precisely, to two significant digits. They intend to survey a sample of tax returns. How large a sample should they take to achieve the desired accuracy with 90% confidence?
• n>17316
• n>8940
• n>15420
• we don’t have enough information to estimate the sample size