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Homework answers / question archive / 1) The sampling distribution of the statistic s12/s22 calculated from samples randomly selected from normally distributed populations, is A) a chi-squared distribution

1) The sampling distribution of the statistic s12/s22 calculated from samples randomly selected from normally distributed populations, is A) a chi-squared distribution

Statistics

1) The sampling distribution of the statistic s12/s22 calculated from samples randomly selected from normally distributed populations, is

A) a chi-squared distribution.

B) a normal distribution.

C) a t distribution.

D) an F distribution.

2) When the population standard deviation is known and the population values are normally distributed, which distribution is used in developing a test about the population mean?

A) Standard normal distribution.

B) Chi-squared distribution.

C) F distribution.

D) t distribution.

3) A sample of n observations was taken from a population. The appropriate chi-squared distribution used for statistical inference about the population variance has

A) n degrees of freedom.

B) n-1 degrees of freedom.

C) n-2 degrees of freedom.

D) n-k degrees of freedom.

4) Which of the following is not necessary to be known in order to compute the p-value?

A) Level of significance.

B) Numerical value of the test statistic.

C) Knowledge of whether the test is upper, lower, or two-tailed.

D) Sampling distribution of the test statistic.

5) Which of the following hypotheses can’t be tested with a chi-square test?

A) Intelligence (IQ score) and height are independently distributed in the human population.

B) 40% of American teenagers have a normal weight, 10% are underweight, 35% are overweight, and 15% are severely obese.

C) The logarithm of income is normally distributed within the population of the United Kingdom.

D) Women in Botswana give birth to their first child on average more than 1 year earlier than they get married.

6) In regression analysis, the explanatory variables are also called as

A) Dependent variables.

B) Independent variables.

C) Parameters.

D) Factors.

7) In regression analysis, the error term is a random variable with an expected value of

A) Any number.

B) One.

C) Any positive number.

D) Zero.

8) A regression model in which more than one explanatory variables are used to predict the dependent variable is called

A) an interactive model.

B) a multiple regression model.

C) an independent model.

D) a multi-dependent model.

9) In hypothesis testing, if the null hypothesis is rejected then

A) no conclusions can be drawn from the test.

B) the alternative hypothesis is accepted as true.

C) the data must have been accumulated incorrectly.

D) the sample provides sufficient evidence against the alternative hypothesis.

10) If a null hypothesis is rejected at the 5% significance level, then

A) it will always be rejected at the 10% significance level.

B) it will never be rejected at the 10% significance level.

C) it will never be accepted at the 1% significance level.

D) it will never be rejected at the 1% significance level.

Task 1    There is a factory in South Mexico producing canned tomato juice. Regarding the legislation the cans have to have  a mean fill volume of at least 250 grams.                  

                a. Formulate hypotheses that could be used to determine whether the mean fill volume satisfies the legal requirement.                    

                b. We took a random sample of 20 cans from the production line of the factory (you can see the results in the G-H columns).  Test your hypothesis using the p value approach. Interpret your results.                       

                c. Test your hypothesis if the significance level=5%. What is your conclusion? Does the factory should adjust the settings of the production line?                        

                Variable of interest         y              Filling volume

                Null hypothesis H0          

ONE-TAILED (LOWER TALE, CL)   Alternative hypothesis  H1          

                hypothesized value        μ0          

                significance level              α            

                sample size         n            

                sample mean     y_bar   

                SAMPLE standard deviation        s             

                standard error   SE(y_bar)           

                degree of freedom         df           

                test statistics                     

                type of test        (LT/UT/2T)         

p-value approach:           p-value                 p            

                decision                              

Critical value approach: critical value                      

                decision                              

                consequence                    

Weights of the cans (grams)

1

230

248

2

241

251

3

256

249

4

244

245

5

266

255

6

234

249

7

248

257

8

254

251

9

229

244

10

252

251

 

Task 2    The police has investigated that whether there is any association between use of safety belt and the outcome of the accidents. The data obtained are shown in the contingency table below. Use 1% significance level and test for the independence of habit using safety belt and the type of injuries. What is your conclusion?                                                                                                                                                                          Points attained Points available

                                                                                                                                                                                                                10

                Observed frequencies (fij)                                                                                                                                                                                         

                                Safety belt used               Safety belt not used       Total                                                                                                                                     

                Minor injuries    512         229         741                                                                                                                                        

                Major injuries    99           125         224                                                                                                                                        

                Fatal injuries      13           22           35                                                                                                                                          

                Total      624         376         1000                                                                                                                                      

                                                                                                                                                                                                               

                Type of test                                                                                                                                                                                       

                Null hypothesis H0                                                                                                                                                                          

                Alternative hypothesis  H1                                                                                                                                                                          

                Number of rows               p                                                                                                                                                                            

                Number of column          q                                                                                                                                                                            

                Significance level              α                                                                                                                                                                            

                                                                                                                                                                                                               

                                                                                                                                                                                                               

                                                                                                                                                                                                               

                Expected frequencies  (eij)                                                                                                                                                                                        

                                Safety belt used               Safety belt not used                                                                                                                                                      

                Minor injuries                                                                                                                                                                                   

                Major injuries                                                                                                                                                                                   

                Fatal injuries                                                                                                                                                                                     

                Total                                                                                                                                                                                     

                                                                                                                                                                                                               

                χ2                                                                                                                                                                                           

                                Safety belt used               Safety belt not used                                                                                                                                                      

                Minor injuries                                                                                                                                                                                   

                Major injuries                                                                                                                                                                                   

                Fatal injuries                                                                                                                                                                                     

                Total                                                                                                                                                                                     

                                                                                                                                                                                                               

                 Test statistic                                                                                                                                                                                     

                Type of test        (LT/UT/2T)                                                                                                                                                                         

                Degree of freedom         DF                                                                                                                                                                          

Critical value approach: Critical value                                                                                                                                                                                      

                Decision                                                                                                                                                                                              

p-value approach:           p-value                 p                                                                                                                                                                            

                Decision                                                                                                                                                                                              

                Conclusion                                                                                                                                                                                         

Task 3    The following statistics is showing the price of flats in Budapest (in EUR) (PRICE) and 2 explanatory variables: area of the flat in nm (AREA) and age of the flat in years (AGE).                                                                                                                                                                                 PRICE    AREA     AGE                                                       Points attained Points available                                                             

                                                                                                                                                                                                                71,000   34                11           1                                                              10                                                          

                a. Develop an estimated regression equation with (AREA) as the independent variable. Determine b0 and b1. Determine the estimated equation.                                                                                                                                                                                                1,00,000               54           22           2                                                                                                                             

                b. Develop an estimated regression equation with both explanatory  variables (AREA and AGE). Determine b0, b1, b2. Determine the estimated equation.                                                                                                                                                                                  2,95,000               120         2              3                                                                                                                             

                c. Interpret the coefficient of the area in the two different models and explain the difference.                                                                                                                                                                                  62,000   38           22           4                                                                                                                             

                d. Compute  R2 and adjusted R2.  Which model is the better?  Why?                                                                                                                                                                                       1,95,000               86           4              5                                                                                                                             

                e. What is the estimated price for a  68 m2 , 10 years old flat by both models?                                                                                                                                                                                     1,11,000               96           25           6                                                                                                                             

                                                                                                                                                                                                                4,88,000                200         3              7                                                                                                                             

a.            b0=                                                        b1=                                                                                                                        3,24,000                133         4              8                                                                                                                             

                Equation                                                                                                                                                                                              1,14,000               50           13           9                                                                                                                             

b.            b0=                                                        b1=                                        b2=                                                                        88,000   46                55           10                                                                                                                          

                Equation                                                                                                                                                                                              1,88,000               88           6              11                                                                                                                          

c.             Interpretation:                                                                                                                                                                                  2,00,000               77           10           12                                                                                                                          

                                                                                                                                                                                                                93,000   44                18           13                                                                                                                          

                                                                                                                                                                                                                1,60,000                92           12           14                                                                                                                          

d.            Simple regression model                                                                              Multiple regression model                                                                                                           1,59,000               77           1              15                                                                                                                          

                                                                                                                                                                                                                                                                                                                                                                                               

                R-square              =                                                             R-square              =                                                                                                                                                                                                                                                                            

                Adj.R-square     =                                                             Adj.R-square     =                                                                                                                                                                                                                                                                            

                                                                                                                                                                                                                                                                                                                                                                                               

                Model selection, explanation                                                                                                                                                                                                                                                                                                                                                                    

                                                                                                                                                                                                                                                                                                                                                                                               

e.            Projection with the simple regression model                                                                      Projection with the multiple regression model                                                                                                                                                                                                                                                                                            

                                                                                                                                                                                                                                                                                                                                                                                               

                Estimated price =                                                             Estimated price =                                                                                                                                                                                                                                                                            

                                                                                                                                                                                                                                                                                                                                                                                               

                                                                                                                                                                                                                                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