Question Set 1: Completing an ANOVA Source Table

A group of researchers wants to measure the difference between the cost of different fruits from six grocery stores in the USA (Walmart, Target, Weis, Schnucks, Trader Joe, and Aldi). Complete the enclosed ANOVA table to help researchers determine the difference between costs. Round all answers to 3 decimal places(except DF).

Source of Variation	Sum of Squares (SS)	Degrees of Freedom (DF)	Mean Square (MS)	F-statistic
Factor (i.e. BetweenGroups)	35.957	5	?	?
Error (i.e. Within Groups)	322.384	?	?
Total	358.341	261

1. df Within Groups (i.e. df Error) = - =
2. MS Between Groups (i.e. MS Factor) = / =
3. MS Within Groups (i.e. MS Error) = / =
4. F statistic = / =

Source	SS	df	MS	F-value
Between Groups	35.957	5	B	D
Within Groups	322.384	A	C
Total	358.341	261

Question Set 1_A:

Use Minitab to compute the p-value for the previous question:

• Choose Graph > Probability Distribution Plot > View Probability
• Click the Distribution dropdown menu > select F
• For 'Numerator degrees of freedom' enter the df for Between Groups (per Question 1)
• For 'Denominator degrees of freedom' enter the df for Within Groups (per Question 1)
• Click Options > select 'A specified x (axis) value'
• For ANOVA, we use 'Right tail'
• For 'X value' enter the F statistic (per Question 1) > OK

Use Question Set 2 to work on Question 2 set from A to K.

Question Set 2: Describing and Comparing Data from Three or More Groups

This question set uses the NHL player salaries datafile Download NHL player salaries datafilewhich can be obtained from Canvas. We want to determine if there are any differences in average NHL player salaries based on their nationality (Canada, USA and Others).

Data Link to NHL Player Salaries datafile: https://1drv.ms/x/s!Aj_pQpiSrV7orCEOJjk8OWVdCHhY?e=AvmNmo

Here is all the data that is in this link:

First Name	Last Name	Born	City	Nationality	Ht	Wt	Salary
Stephen	Gionta	10/9/1983	Rochester	USA	67	175	575000
Sergey	Tolchinsky	2/3/1995	Moscow	Other	68	170	575000
T.J.	Tynan	2/25/1992	Orland Park	USA	68	165	600000
Viktor	Arvidsson	4/8/1993	Skellefte	Other	69	180	640000
Ryan	Callahan	3/21/1985	Rochester	USA	70	187	6500000
Yannick	Weber	9/23/1988	Morges	Other	71	200	575000
Joel	Vermin	2/5/1992	Bern	Other	71	192	575000
Miikka	Salomaki	3/9/1993	Raahe	Other	71	203	575000
Kevin	Porter	3/12/1986	Detroit	USA	71	191	575000
Patrick	Cannone	8/9/1986	Bayport	USA	71	197	600000
Derek	Ryan	12/29/1986	Spokane	USA	71	170	600000
Dominik	Simon	8/8/1994	Prague	Other	71	176	667500
Nikita	Nesterov	3/28/1993	Chelyabinsk	Other	71	191	725000
Nikita	Soshnikov	10/14/1993	Nizhny Tagil	Other	71	190	742500
Griffen	Molino	1/21/1994	TRENTON	USA	71	171	925000
Jeff	Skinner	5/16/1992	Toronto	CAN	71	200	6000000
Joe	Pavelski	7/11/1984	Plover	USA	71	190	6000000
Tomas	Plekanec	10/31/1982	Kladno	Other	71	196	7000000
Zach	Parise	7/28/1984	Minneapolis	USA	71	196	9000000
Sidney	Crosby	8/7/1987	Cole Harbour	CAN	71	200	10900000
Patrick	Kane	11/19/1988	Buffalo	USA	71	177	13800000
Steve	Oleksy	2/4/1986	Chesterfield	USA	72	190	575000
Oliver	Bjorkstrand	4/10/1995	Herning	Other	72	175	600000
Lukas	Sedlak	2/25/1993	Cesk_ Budejovice	Other	72	203	600000
Ryan	Carpenter	1/18/1991	Oviedo	USA	72	195	600000
Nick	Jensen	9/21/1990	St. Paul	USA	72	196	600000
Pierre-Edouard	Bellemare	3/6/1985	Le Blanc-Mesnil	Other	72	198	725000
Ivan	Barbashev	12/14/1995	Moscow	Other	72	180	742500
Markus	Granlund	4/16/1993	Oulu	Other	72	183	850000
Phillip	Di Giuseppe	10/9/1993	Toronto	CAN	72	200	867500
Brandon	Tanev	12/31/1991	Toronto	CAN	72	180	874125
Chris	Kelly	11/11/1980	Toronto	CAN	72	190	900000
Dominic	Moore	8/3/1980	Thornhill	CAN	72	192	900000
Nick	Schmaltz	2/23/1996	Madison	USA	72	177	925000
J.T.	Compher	4/8/1995	Northbrook	USA	72	193	925000
Michael	Kapla	9/19/1994	Eau Claire	USA	72	200	925000
Justin	Faulk	3/20/1992	South St.Paul	USA	72	215	5500000
Dennis	Wideman	3/20/1983	Kitchener	CAN	72	202	6000000
Alex	Steen	3/1/1984	Winnipeg	Other	72	211	6500000
Mark	Giordano	10/3/1983	Toronto	CAN	72	198	6750000
Paul	Stastny	12/27/1985	Quebec City	USA	72	202	7000000
Erik	Karlsson	5/31/1990	Landsbro	Other	72	191	7000000
Kris	Letang	4/24/1987	Montreal	CAN	72	201	7250000
Vladimir	Tarasenko	12/13/1991	Yaroslavl	Other	72	219	8000000
Kyle	Okposo	4/16/1988	St. Paul	USA	72	218	8000000
Phil	Kessel	10/2/1987	Madison	USA	72	202	9000000
P.K.	Subban	5/13/1989	Toronto	CAN	72	210	11000000
Matt	Irwin	11/29/1987	Victoria	CAN	73	207	575000
Paul	Carey	9/24/1988	Boston	USA	73	198	575000
Michal	Rozsival	9/3/1978	Vlasim	Other	73	210	600000
Nick	Shore	9/26/1992	Denver	USA	73	201	600000
Martin	Frk	10/5/1993	Pelhrimov	Other	73	194	650000
Fredrik	Claesson	11/24/1992	Stockholm	Other	73	200	700000
Oscar	Lindberg	10/29/1991	Skellefte	Other	73	202	700000
Xavier	Ouellet	7/29/1993	Bayonne	Other	73	208	715000
Valentin	Zykov	5/15/1995	St. Petersburg	Other	73	224	727500
Phillip	Danault	2/24/1993	Victoriaville	CAN	73	193	875000
Nate	Schmidt	7/16/1991	St. Cloud	USA	73	194	875000
Colin	Miller	10/29/1992	Sault Ste. Marie	CAN	73	196	900000
Adam	Erne	4/20/1995	New Haven	USA	73	214	925000
Brock	Boeser	2/25/1997	Burnsville	USA	73	191	925000
Jason	Demers	6/9/1988	Dorval	CAN	73	200	5500000
Brandon	Saad	10/27/1992	Pittsburgh	USA	73	206	5750000
Matt	Niskanen	12/6/1986	Virginia	USA	73	200	5750000
Tyler	Seguin	1/31/1992	Brampton	CAN	73	200	6000000
Taylor	Hall	11/14/1991	Calgary	CAN	73	201	6000000
Marian	Gaborik	2/14/1982	Trencin	Other	73	205	6075000
Nicklas	Backstrom	11/23/1987	G_vle	Other	73	213	7000000
Daniel	Sedin	9/26/1980	Ornskoldsvik	Other	73	190	7000000
Drew	Doughty	12/8/1989	London	CAN	73	195	7450000
Patrice	Bergeron	7/24/1985	Ancienne-Lorette	CAN	73	195	8750000
Borna	Rendulic	3/25/1992	Zagreb	Other	74	200	575000
Tim	Schaller	11/16/1990	Merrimack	USA	74	219	600000
Barclay	Goodrow	2/26/1993	Toronto	CAN	74	215	635000
Peter	Cehlarik	8/2/1995	Zilina	Other	74	202	692500
Markus	Hannikainen	3/26/1993	Helsinki	Other	74	195	742500
Matthew	Tkachuk	12/11/1997	Scottsdale	USA	74	202	925000
Mike	Reilly	7/13/1993	Chicago	USA	74	193	925000
Zach	Werenski	7/19/1997	Grosse Pointe	USA	74	218	925000
Justin	Abdelkader	2/25/1987	Muskegon	USA	74	218	5500000
Jamie	Benn	7/18/1989	Victoria	CAN	74	210	5750000
Brandon	Dubinsky	4/29/1986	Anchorage	USA	74	218	5850000
Patrick	Marleau	9/15/1979	Aneroid	CAN	74	215	6500000
Johnny	Boychuk	1/19/1984	Edmonton	CAN	74	227	7000000
Ryan	Kesler	8/31/1984	Livonia	USA	74	202	7875000
Ryan	Suter	1/21/1985	Madison	USA	74	206	9000000
Jakub	Voracek	8/15/1989	Kladno	Other	74	214	10000000
Jonathan	Toews	4/29/1988	Winnipeg	CAN	74	201	13800000
Dennis	Rasmussen	7/3/1990	Vasteras	Other	75	205	575000
Tomas	Nosek	9/1/1992	Pardubice	Other	75	210	575000
Andreas	Martinsen	6/13/1990	B_rum	Other	75	220	640000
Petteri	Lindbohm	9/23/1993	Helsinki	Other	75	198	645000
Magnus	Paajarvi	4/12/1991	Norrkoping	Other	75	203	700000
Korbinian	Holzer	2/16/1988	Munich	Other	75	215	700000
Tyler	Gaudet	4/4/1993	Hamilton	CAN	75	205	874125
Duncan	Siemens	9/7/1993	Edmonton	CAN	75	210	874125
Brett	Ritchie	7/1/1993	Orangeville	CAN	75	215	875000
Joel	Armia	5/31/1993	Pori	Other	75	205	875000
Jakob	Chychrun	3/31/1998	Boca Raton	USA	75	200	925000
Auston	Matthews	9/17/1997	Scottsdale	USA	75	216	925000
Brooks	Orpik	9/26/1980	San Francisco	USA	75	219	5500000
Ryan	Johansen	7/31/1992	Vancouver	CAN	75	218	6000000
Jeff	Carter	1/1/1985	London	CAN	75	217	7000000
Alex	Pietrangelo	1/18/1990	King City	CAN	75	205	7000000
Jeff	Petry	12/9/1987	Ann Arbor	USA	75	204	7000000
Dion	Phaneuf	4/10/1985	Edmonton	CAN	75	219	7500000
Milan	Lucic	6/7/1988	Vancouver	CAN	75	236	8000000
Andrew	Ladd	12/12/1985	Maple Ridge	CAN	75	200	8000000
David	Backes	5/1/1984	Minneapolis	USA	75	221	8000000
Evgeni	Malkin	7/31/1986	Magnitogorsk	Other	75	195	9500000
Alex	Ovechkin	9/17/1985	Moscow	Other	75	239	10000000
Scott	Mayfield	10/14/1992	St. Louis	USA	76	224	575000
Zach	Sanford	11/9/1994	Salem	USA	76	203	875000
Joel	Edmundson	6/28/1993	Brandon	CAN	76	220	900000
Alex	Tuch	5/10/1996	Syracuse	USA	76	222	925000
Jay	Bouwmeester	9/27/1983	Edmonton	CAN	76	221	5600000
Artem	Anisimov	5/24/1988	Yaroslavl	Other	76	198	5750000
Erik	Johnson	3/21/1988	Bloomington	USA	76	225	6000000
Joe	Thornton	7/2/1979	London	CAN	76	220	6750000
Rick	Nash	6/16/1984	Brampton	CAN	76	212	8000000
Shea	Weber	8/14/1985	Sicamous	CAN	76	232	12000000
Blake	Wheeler	8/31/1986	Plymouth	USA	77	225	5800000

Question Set 2_A:

Use Minitab to find the number of cases, mean and standard deviation for the salary variable for each of the three nationalities separately:

• Choose Stat > Basic Statistics > Display Descriptive Statistics
• Variable: double-click on 'Salary'
• Group variables: double-click on 'Nationality'
• Click Statistics > select Mean, Standard deviation and N nonmissing > OK

Question Set 2_B:

Use Minitab to construct side-by-side boxplots to compare NHL players concerning their nationality and salaries:

• Choose Graph > Boxplot > One Y Variable, With Categorical Variables
• For Y-variable, double-click on 'Salary'
• For Categorical variables, double-click on 'Nationality' > OK

Question Set 2_C:

Use Minitab to construct a dotplot with groups to compare NHL players with respect to their nationality and salaries:

• Choose Graph > Dotplot > One Y Variable, Groups Displayed on the Scale
• For Y-variable, double-click on 'Salary'
• For Group variables, double-click on 'Nationality' > OK

Question Set 2_D:

Explain why a one-way ANOVA should be conducted to compare the mean salary of the players nationality groups as opposed to a series of three independent means t tests.

Question Set 2_E:

State the null and alternative hypotheses. (You will check the assumptions in the next question.)

Question Set 2_F:

Check the assumptions:

The first ANOVA assumption is that the samples are independent. The other assumptions include: The response variable for each factor level needs to be approximately normally distributed OR the sample size for each factor level needs to be at least 30. Is this assumption met (per the summary statistics you computed above)? Explain your answer.

The population variances need to be equal for the three Nationality groups. Using the summary statistics, if the largest standard deviation divided by the smallest standard deviation is less than 2, then we can assume that the population variances are equal. Is this assumption met? Show your work.

Question Set 2_G:

Compute the test statistic and the p-value.

Specifically, use Minitab to conduct a one-way ANOVA to compare the mean Salary for the three Nationality groups:

• Choose Stat > ANOVA > One-Way
• For Response, double-click on 'Salary'
• For Factor, double-click on 'Nationality' > OK

Per your Minitab results, what is the test statistic value (i.e. F-Value)? Use 2 decimal places

Question Set 2_H:

Decide to reject or fail to reject the null hypothesis.Specifically, fill in the following blanks:

Because the p-value is _______ (use 3 decimal places), which is _____________ (less than or greater than) our alpha-level of 0.05, we _____________ (reject or fail to reject) the null hypothesis.

Question Set 2_I:

State a real-world conclusion. Specifically, is there statistical evidence that the population mean salary for all three nationalities are NOT all equal? Explain your answer.

Question Set 2_J:

We have concluded above that not all population means are equal. The next step is to determine which population means are different. Therefore, use Minitab to conduct Tukey simultaneous tests for differences in means:

• Choose Stat > ANOVA > One-Way
• Your Response and Factor should already be entered from before
• Click Comparisons > select 'Tukey' comparisons > OK

Image to provide a screenshot of your Minitab "Tukey Simultaneous 95% CIs" graph.

Question Set 2_K:

Given your Tukey comparisons results above, which pairs of population means are significantly different? Specifically, answer these three questions:

1. Is the mean salary for NHL Players from Canada significantly different than the mean salary for players from Other?
2. Is the mean Salary for NHL player from Canada significantly different than the mean Salary for NHL player from the USA?
3. Is the mean Salary for NHL player from Other significantly different than the mean Salary for NHL player from the USA?