Benjamin Harris – Assignment 3 – The Study of Wine Quality Part C

Requirements for the Assignment

Pre-Requisites to answering the questions – Load in the Red Wine Dataset and Preview top 100 rows

wine <- read.csv(file = "winequality-red.csv", sep=";", header = T) # Load in the dataset
knitr::kable(head(wine,100), caption = "Red Wine Dataset")

Red Wine Dataset

fixed.acidity	volatile.acidity	citric.acid	residual.sugar	chlorides	free.sulfur.dioxide	total.sulfur.dioxide	density	pH	sulphates	alcohol	quality
7.4	0.700	0.00	1.90	0.076	11	34	0.9978	3.51	0.56	9.4	5
7.8	0.880	0.00	2.60	0.098	25	67	0.9968	3.20	0.68	9.8	5
7.8	0.760	0.04	2.30	0.092	15	54	0.9970	3.26	0.65	9.8	5
11.2	0.280	0.56	1.90	0.075	17	60	0.9980	3.16	0.58	9.8	6
7.4	0.700	0.00	1.90	0.076	11	34	0.9978	3.51	0.56	9.4	5
7.4	0.660	0.00	1.80	0.075	13	40	0.9978	3.51	0.56	9.4	5
7.9	0.600	0.06	1.60	0.069	15	59	0.9964	3.30	0.46	9.4	5
7.3	0.650	0.00	1.20	0.065	15	21	0.9946	3.39	0.47	10.0	7
7.8	0.580	0.02	2.00	0.073	9	18	0.9968	3.36	0.57	9.5	7
7.5	0.500	0.36	6.10	0.071	17	102	0.9978	3.35	0.80	10.5	5
6.7	0.580	0.08	1.80	0.097	15	65	0.9959	3.28	0.54	9.2	5
7.5	0.500	0.36	6.10	0.071	17	102	0.9978	3.35	0.80	10.5	5
5.6	0.615	0.00	1.60	0.089	16	59	0.9943	3.58	0.52	9.9	5
7.8	0.610	0.29	1.60	0.114	9	29	0.9974	3.26	1.56	9.1	5
8.9	0.620	0.18	3.80	0.176	52	145	0.9986	3.16	0.88	9.2	5
8.9	0.620	0.19	3.90	0.170	51	148	0.9986	3.17	0.93	9.2	5
8.5	0.280	0.56	1.80	0.092	35	103	0.9969	3.30	0.75	10.5	7
8.1	0.560	0.28	1.70	0.368	16	56	0.9968	3.11	1.28	9.3	5
7.4	0.590	0.08	4.40	0.086	6	29	0.9974	3.38	0.50	9.0	4
7.9	0.320	0.51	1.80	0.341	17	56	0.9969	3.04	1.08	9.2	6
8.9	0.220	0.48	1.80	0.077	29	60	0.9968	3.39	0.53	9.4	6
7.6	0.390	0.31	2.30	0.082	23	71	0.9982	3.52	0.65	9.7	5
7.9	0.430	0.21	1.60	0.106	10	37	0.9966	3.17	0.91	9.5	5
8.5	0.490	0.11	2.30	0.084	9	67	0.9968	3.17	0.53	9.4	5
6.9	0.400	0.14	2.40	0.085	21	40	0.9968	3.43	0.63	9.7	6
6.3	0.390	0.16	1.40	0.080	11	23	0.9955	3.34	0.56	9.3	5
7.6	0.410	0.24	1.80	0.080	4	11	0.9962	3.28	0.59	9.5	5
7.9	0.430	0.21	1.60	0.106	10	37	0.9966	3.17	0.91	9.5	5
7.1	0.710	0.00	1.90	0.080	14	35	0.9972	3.47	0.55	9.4	5
7.8	0.645	0.00	2.00	0.082	8	16	0.9964	3.38	0.59	9.8	6
6.7	0.675	0.07	2.40	0.089	17	82	0.9958	3.35	0.54	10.1	5
6.9	0.685	0.00	2.50	0.105	22	37	0.9966	3.46	0.57	10.6	6
8.3	0.655	0.12	2.30	0.083	15	113	0.9966	3.17	0.66	9.8	5
6.9	0.605	0.12	10.70	0.073	40	83	0.9993	3.45	0.52	9.4	6
5.2	0.320	0.25	1.80	0.103	13	50	0.9957	3.38	0.55	9.2	5
7.8	0.645	0.00	5.50	0.086	5	18	0.9986	3.40	0.55	9.6	6
7.8	0.600	0.14	2.40	0.086	3	15	0.9975	3.42	0.60	10.8	6
8.1	0.380	0.28	2.10	0.066	13	30	0.9968	3.23	0.73	9.7	7
5.7	1.130	0.09	1.50	0.172	7	19	0.9940	3.50	0.48	9.8	4
7.3	0.450	0.36	5.90	0.074	12	87	0.9978	3.33	0.83	10.5	5
7.3	0.450	0.36	5.90	0.074	12	87	0.9978	3.33	0.83	10.5	5
8.8	0.610	0.30	2.80	0.088	17	46	0.9976	3.26	0.51	9.3	4
7.5	0.490	0.20	2.60	0.332	8	14	0.9968	3.21	0.90	10.5	6
8.1	0.660	0.22	2.20	0.069	9	23	0.9968	3.30	1.20	10.3	5
6.8	0.670	0.02	1.80	0.050	5	11	0.9962	3.48	0.52	9.5	5
4.6	0.520	0.15	2.10	0.054	8	65	0.9934	3.90	0.56	13.1	4
7.7	0.935	0.43	2.20	0.114	22	114	0.9970	3.25	0.73	9.2	5
8.7	0.290	0.52	1.60	0.113	12	37	0.9969	3.25	0.58	9.5	5
6.4	0.400	0.23	1.60	0.066	5	12	0.9958	3.34	0.56	9.2	5
5.6	0.310	0.37	1.40	0.074	12	96	0.9954	3.32	0.58	9.2	5
8.8	0.660	0.26	1.70	0.074	4	23	0.9971	3.15	0.74	9.2	5
6.6	0.520	0.04	2.20	0.069	8	15	0.9956	3.40	0.63	9.4	6
6.6	0.500	0.04	2.10	0.068	6	14	0.9955	3.39	0.64	9.4	6
8.6	0.380	0.36	3.00	0.081	30	119	0.9970	3.20	0.56	9.4	5
7.6	0.510	0.15	2.80	0.110	33	73	0.9955	3.17	0.63	10.2	6
7.7	0.620	0.04	3.80	0.084	25	45	0.9978	3.34	0.53	9.5	5
10.2	0.420	0.57	3.40	0.070	4	10	0.9971	3.04	0.63	9.6	5
7.5	0.630	0.12	5.10	0.111	50	110	0.9983	3.26	0.77	9.4	5
7.8	0.590	0.18	2.30	0.076	17	54	0.9975	3.43	0.59	10.0	5
7.3	0.390	0.31	2.40	0.074	9	46	0.9962	3.41	0.54	9.4	6
8.8	0.400	0.40	2.20	0.079	19	52	0.9980	3.44	0.64	9.2	5
7.7	0.690	0.49	1.80	0.115	20	112	0.9968	3.21	0.71	9.3	5
7.5	0.520	0.16	1.90	0.085	12	35	0.9968	3.38	0.62	9.5	7
7.0	0.735	0.05	2.00	0.081	13	54	0.9966	3.39	0.57	9.8	5
7.2	0.725	0.05	4.65	0.086	4	11	0.9962	3.41	0.39	10.9	5
7.2	0.725	0.05	4.65	0.086	4	11	0.9962	3.41	0.39	10.9	5
7.5	0.520	0.11	1.50	0.079	11	39	0.9968	3.42	0.58	9.6	5
6.6	0.705	0.07	1.60	0.076	6	15	0.9962	3.44	0.58	10.7	5
9.3	0.320	0.57	2.00	0.074	27	65	0.9969	3.28	0.79	10.7	5
8.0	0.705	0.05	1.90	0.074	8	19	0.9962	3.34	0.95	10.5	6
7.7	0.630	0.08	1.90	0.076	15	27	0.9967	3.32	0.54	9.5	6
7.7	0.670	0.23	2.10	0.088	17	96	0.9962	3.32	0.48	9.5	5
7.7	0.690	0.22	1.90	0.084	18	94	0.9961	3.31	0.48	9.5	5
8.3	0.675	0.26	2.10	0.084	11	43	0.9976	3.31	0.53	9.2	4
9.7	0.320	0.54	2.50	0.094	28	83	0.9984	3.28	0.82	9.6	5
8.8	0.410	0.64	2.20	0.093	9	42	0.9986	3.54	0.66	10.5	5
8.8	0.410	0.64	2.20	0.093	9	42	0.9986	3.54	0.66	10.5	5
6.8	0.785	0.00	2.40	0.104	14	30	0.9966	3.52	0.55	10.7	6
6.7	0.750	0.12	2.00	0.086	12	80	0.9958	3.38	0.52	10.1	5
8.3	0.625	0.20	1.50	0.080	27	119	0.9972	3.16	1.12	9.1	4
6.2	0.450	0.20	1.60	0.069	3	15	0.9958	3.41	0.56	9.2	5
7.8	0.430	0.70	1.90	0.464	22	67	0.9974	3.13	1.28	9.4	5
7.4	0.500	0.47	2.00	0.086	21	73	0.9970	3.36	0.57	9.1	5
7.3	0.670	0.26	1.80	0.401	16	51	0.9969	3.16	1.14	9.4	5
6.3	0.300	0.48	1.80	0.069	18	61	0.9959	3.44	0.78	10.3	6
6.9	0.550	0.15	2.20	0.076	19	40	0.9961	3.41	0.59	10.1	5
8.6	0.490	0.28	1.90	0.110	20	136	0.9972	2.93	1.95	9.9	6
7.7	0.490	0.26	1.90	0.062	9	31	0.9966	3.39	0.64	9.6	5
9.3	0.390	0.44	2.10	0.107	34	125	0.9978	3.14	1.22	9.5	5
7.0	0.620	0.08	1.80	0.076	8	24	0.9978	3.48	0.53	9.0	5
7.9	0.520	0.26	1.90	0.079	42	140	0.9964	3.23	0.54	9.5	5
8.6	0.490	0.28	1.90	0.110	20	136	0.9972	2.93	1.95	9.9	6
8.6	0.490	0.29	2.00	0.110	19	133	0.9972	2.93	1.98	9.8	5
7.7	0.490	0.26	1.90	0.062	9	31	0.9966	3.39	0.64	9.6	5
5.0	1.020	0.04	1.40	0.045	41	85	0.9938	3.75	0.48	10.5	4
4.7	0.600	0.17	2.30	0.058	17	106	0.9932	3.85	0.60	12.9	6
6.8	0.775	0.00	3.00	0.102	8	23	0.9965	3.45	0.56	10.7	5
7.0	0.500	0.25	2.00	0.070	3	22	0.9963	3.25	0.63	9.2	5
7.6	0.900	0.06	2.50	0.079	5	10	0.9967	3.39	0.56	9.8	5
8.1	0.545	0.18	1.90	0.080	13	35	0.9972	3.30	0.59	9.0	6

1. Produce summary statistics of “residual.sugar” and use its median to divide the data into two groups A and B. We want to test if “density” in Group A and Group B has the same population mean. Please answer the following questions.

Pre-requisites to answering the questions – provide summary statistics of “residual.sugar” and use median to divide the data into two groups A and B

# Summary Statistics
summary(wine$residual.sugar)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   0.900   1.900   2.200   2.539   2.600  15.500

# Use the median (2.2) to divide the data into two groups A and B
rs.group <- ifelse(wine$residual.sugar < 2.2, "A", "B")
split_df <- data.frame(wine$residual.sugar, rs.group, wine$density)

# show the data after division (top 100 rows)
print(head(split_df, 100))

##     wine.residual.sugar rs.group wine.density
## 1                  1.90        A       0.9978
## 2                  2.60        B       0.9968
## 3                  2.30        B       0.9970
## 4                  1.90        A       0.9980
## 5                  1.90        A       0.9978
## 6                  1.80        A       0.9978
## 7                  1.60        A       0.9964
## 8                  1.20        A       0.9946
## 9                  2.00        A       0.9968
## 10                 6.10        B       0.9978
## 11                 1.80        A       0.9959
## 12                 6.10        B       0.9978
## 13                 1.60        A       0.9943
## 14                 1.60        A       0.9974
## 15                 3.80        B       0.9986
## 16                 3.90        B       0.9986
## 17                 1.80        A       0.9969
## 18                 1.70        A       0.9968
## 19                 4.40        B       0.9974
## 20                 1.80        A       0.9969
## 21                 1.80        A       0.9968
## 22                 2.30        B       0.9982
## 23                 1.60        A       0.9966
## 24                 2.30        B       0.9968
## 25                 2.40        B       0.9968
## 26                 1.40        A       0.9955
## 27                 1.80        A       0.9962
## 28                 1.60        A       0.9966
## 29                 1.90        A       0.9972
## 30                 2.00        A       0.9964
## 31                 2.40        B       0.9958
## 32                 2.50        B       0.9966
## 33                 2.30        B       0.9966
## 34                10.70        B       0.9993
## 35                 1.80        A       0.9957
## 36                 5.50        B       0.9986
## 37                 2.40        B       0.9975
## 38                 2.10        A       0.9968
## 39                 1.50        A       0.9940
## 40                 5.90        B       0.9978
## 41                 5.90        B       0.9978
## 42                 2.80        B       0.9976
## 43                 2.60        B       0.9968
## 44                 2.20        B       0.9968
## 45                 1.80        A       0.9962
## 46                 2.10        A       0.9934
## 47                 2.20        B       0.9970
## 48                 1.60        A       0.9969
## 49                 1.60        A       0.9958
## 50                 1.40        A       0.9954
## 51                 1.70        A       0.9971
## 52                 2.20        B       0.9956
## 53                 2.10        A       0.9955
## 54                 3.00        B       0.9970
## 55                 2.80        B       0.9955
## 56                 3.80        B       0.9978
## 57                 3.40        B       0.9971
## 58                 5.10        B       0.9983
## 59                 2.30        B       0.9975
## 60                 2.40        B       0.9962
## 61                 2.20        B       0.9980
## 62                 1.80        A       0.9968
## 63                 1.90        A       0.9968
## 64                 2.00        A       0.9966
## 65                 4.65        B       0.9962
## 66                 4.65        B       0.9962
## 67                 1.50        A       0.9968
## 68                 1.60        A       0.9962
## 69                 2.00        A       0.9969
## 70                 1.90        A       0.9962
## 71                 1.90        A       0.9967
## 72                 2.10        A       0.9962
## 73                 1.90        A       0.9961
## 74                 2.10        A       0.9976
## 75                 2.50        B       0.9984
## 76                 2.20        B       0.9986
## 77                 2.20        B       0.9986
## 78                 2.40        B       0.9966
## 79                 2.00        A       0.9958
## 80                 1.50        A       0.9972
## 81                 1.60        A       0.9958
## 82                 1.90        A       0.9974
## 83                 2.00        A       0.9970
## 84                 1.80        A       0.9969
## 85                 1.80        A       0.9959
## 86                 2.20        B       0.9961
## 87                 1.90        A       0.9972
## 88                 1.90        A       0.9966
## 89                 2.10        A       0.9978
## 90                 1.80        A       0.9978
## 91                 1.90        A       0.9964
## 92                 1.90        A       0.9972
## 93                 2.00        A       0.9972
## 94                 1.90        A       0.9966
## 95                 1.40        A       0.9938
## 96                 2.30        B       0.9932
## 97                 3.00        B       0.9965
## 98                 2.00        A       0.9963
## 99                 2.50        B       0.9967
## 100                1.90        A       0.9972

a. State the null Hypothesis

• There is no difference in the “density” means between groups A and B

b. Use visualization tools to inspect the hypothesis. Do you think the hypothesis is right or not?

boxplot(wine$density ~ rs.group)

• Given the boxplots in the output above, I think the null hypothesis will get rejected; I think there is a difference in means between the groups

c. What test are you going to use?

• I am going to use a t-test

d. What is the p-value?

t.test(wine$density ~ rs.group)

## 
##  Welch Two Sample t-test
## 
## data:  wine$density by rs.group
## t = -14.955, df = 1571.9, p-value < 2.2e-16
## alternative hypothesis: true difference in means between group A and group B is not equal to 0
## 95 percent confidence interval:
##  -0.001479826 -0.001136653
## sample estimates:
## mean in group A mean in group B 
##       0.9960537       0.9973619

• As can be seen in the above code output, the p-value is less than 2.2e-16

e. What is your conclusion?

• There is a difference in “density” means between groups A and B

f. Does your conclusion imply that there is an association between “density” and “residual.sugar”?

• Yes, the conclusion implies that there is an association between the “density” and “residual.sugar”

2. Produce summary statistics of “residual.sugar” and use its 1st, 2nd, and 3rd quantiles to divide the data into four groups A, B, C, and D. We want to test if “density” in the four groups has the same population mean. Please answer the following questions.

Pre-requisites to answering the questions (provide summary statistics of “residual.sugar” and use 1st, 2nd, and 3rd quartiles to divide the data into four groups A, B, C, and D)

# Summary Statistics
summary(wine$residual.sugar)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   0.900   1.900   2.200   2.539   2.600  15.500

# split the groups up using the 1st, 2nd, and 3rd quartiles
rs.group2 <- NULL
for (i in 1:length(wine$residual.sugar)){
  if(wine$residual.sugar[i] <= 1.9) rs.group2[i] <- "A"
    else if(wine$residual.sugar[i] <= 2.2) rs.group2[i] <- "B"
      else if(wine$residual.sugar[i] <= 2.6) rs.group2[i] <- "C"
        else rs.group2[i] <- "D"
}

# preview counts in each group
table(rs.group2)

## rs.group2
##   A   B   C   D 
## 464 419 361 355

# put relevant columns into dataframe and preview top 100 rows
head(data.frame(wine$residual.sugar, rs.group2, wine$density), 100)

##     wine.residual.sugar rs.group2 wine.density
## 1                  1.90         A       0.9978
## 2                  2.60         C       0.9968
## 3                  2.30         C       0.9970
## 4                  1.90         A       0.9980
## 5                  1.90         A       0.9978
## 6                  1.80         A       0.9978
## 7                  1.60         A       0.9964
## 8                  1.20         A       0.9946
## 9                  2.00         B       0.9968
## 10                 6.10         D       0.9978
## 11                 1.80         A       0.9959
## 12                 6.10         D       0.9978
## 13                 1.60         A       0.9943
## 14                 1.60         A       0.9974
## 15                 3.80         D       0.9986
## 16                 3.90         D       0.9986
## 17                 1.80         A       0.9969
## 18                 1.70         A       0.9968
## 19                 4.40         D       0.9974
## 20                 1.80         A       0.9969
## 21                 1.80         A       0.9968
## 22                 2.30         C       0.9982
## 23                 1.60         A       0.9966
## 24                 2.30         C       0.9968
## 25                 2.40         C       0.9968
## 26                 1.40         A       0.9955
## 27                 1.80         A       0.9962
## 28                 1.60         A       0.9966
## 29                 1.90         A       0.9972
## 30                 2.00         B       0.9964
## 31                 2.40         C       0.9958
## 32                 2.50         C       0.9966
## 33                 2.30         C       0.9966
## 34                10.70         D       0.9993
## 35                 1.80         A       0.9957
## 36                 5.50         D       0.9986
## 37                 2.40         C       0.9975
## 38                 2.10         B       0.9968
## 39                 1.50         A       0.9940
## 40                 5.90         D       0.9978
## 41                 5.90         D       0.9978
## 42                 2.80         D       0.9976
## 43                 2.60         C       0.9968
## 44                 2.20         B       0.9968
## 45                 1.80         A       0.9962
## 46                 2.10         B       0.9934
## 47                 2.20         B       0.9970
## 48                 1.60         A       0.9969
## 49                 1.60         A       0.9958
## 50                 1.40         A       0.9954
## 51                 1.70         A       0.9971
## 52                 2.20         B       0.9956
## 53                 2.10         B       0.9955
## 54                 3.00         D       0.9970
## 55                 2.80         D       0.9955
## 56                 3.80         D       0.9978
## 57                 3.40         D       0.9971
## 58                 5.10         D       0.9983
## 59                 2.30         C       0.9975
## 60                 2.40         C       0.9962
## 61                 2.20         B       0.9980
## 62                 1.80         A       0.9968
## 63                 1.90         A       0.9968
## 64                 2.00         B       0.9966
## 65                 4.65         D       0.9962
## 66                 4.65         D       0.9962
## 67                 1.50         A       0.9968
## 68                 1.60         A       0.9962
## 69                 2.00         B       0.9969
## 70                 1.90         A       0.9962
## 71                 1.90         A       0.9967
## 72                 2.10         B       0.9962
## 73                 1.90         A       0.9961
## 74                 2.10         B       0.9976
## 75                 2.50         C       0.9984
## 76                 2.20         B       0.9986
## 77                 2.20         B       0.9986
## 78                 2.40         C       0.9966
## 79                 2.00         B       0.9958
## 80                 1.50         A       0.9972
## 81                 1.60         A       0.9958
## 82                 1.90         A       0.9974
## 83                 2.00         B       0.9970
## 84                 1.80         A       0.9969
## 85                 1.80         A       0.9959
## 86                 2.20         B       0.9961
## 87                 1.90         A       0.9972
## 88                 1.90         A       0.9966
## 89                 2.10         B       0.9978
## 90                 1.80         A       0.9978
## 91                 1.90         A       0.9964
## 92                 1.90         A       0.9972
## 93                 2.00         B       0.9972
## 94                 1.90         A       0.9966
## 95                 1.40         A       0.9938
## 96                 2.30         C       0.9932
## 97                 3.00         D       0.9965
## 98                 2.00         B       0.9963
## 99                 2.50         C       0.9967
## 100                1.90         A       0.9972

a. State the null Hypothesis

• No difference in the “density” means between groups A, B, C, and D

b. Use visualization tools to inspect the hypothesis. Do you think the hypothesis is right or not?

boxplot(wine$density ~ rs.group2)

• Given the boxplots in the output above, I think the null hypothesis will get rejected; I think there is a difference in means between the groups

c. What test are you going to use?

• I am going to use an ANOVA test

d. What is the p-value?

summary(aov(wine$density ~ rs.group2))

##               Df   Sum Sq   Mean Sq F value Pr(>F)    
## rs.group2      3 0.000996 0.0003321   112.8 <2e-16 ***
## Residuals   1595 0.004696 0.0000029                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

• As can be seen in the above code output, the p-value is less than 2e-16

e. What is your conclusion?

• At least one of the “density” group means is different from the rest

f. Does your conclusion imply that there is an association between “density” and “residual.sugar”? Compare your result here with that in Question 1. Do you think increasing the number of groups help identify the association? Would you consider dividing the data into 10 groups so as to help the discovery of the association? Why?

• The implication is that there is an association between “density” and “residual.sugar”
• This p-value is less than that of the two-sample t-test; per the outputs, this p-value < 2e-16 and the t-test p-value < 2.2e-16
• Provided the lower p-value, I think increasing number of groups helps identify the association. Additionally, dividing the data up into more groups presents more opportunities for identifying differences in statistical aggregate measures (such as the mean)
• In this case, I do not think dividing the data up into ten groups for discovery of association is warranted or would be helpful; the p-values of this test and the last one were extremely small. However, there are other instances/applications where dividing the data into 10 groups would be warranted and furthermore, would help with the discovery of association

3. Create a 2 by 4 contingency table using the categories A, B, C, D of “residual.sugar” and the binary variable “excellent” you created in Part B. Note that you have two factors: the categorical levels of “residual.sugar” (A, B, C and D) and an indicator of excellent wines (yes or no).

Re-create the “excellent” variable with “Yes” and “No” Values

wine$excellent <- ifelse(wine$quality >= 7, "Yes", "No")

Create the contingency table

# create it
ctable <- table(data.frame(wine$excellent, rs.group2))

# preview it
print(ctable)

##               rs.group2
## wine.excellent   A   B   C   D
##            No  411 367 308 296
##            Yes  53  52  53  59

a. Use the Chi-square test to test if these two factors are correlated or not

chisq.test(ctable)

## 
##  Pearson's Chi-squared test
## 
## data:  ctable
## X-squared = 5.5, df = 3, p-value = 0.1386

• According to the results of the above chi-squared test, there is no association/correlation (p > 0.05) between wine excellence and residual sugar

b. Use the permutation test to do the same and compare the result to that in (a)

chisq.test(ctable, simulate.p.value = T)

## 
##  Pearson's Chi-squared test with simulated p-value (based on 2000
##  replicates)
## 
## data:  ctable
## X-squared = 5.5, df = NA, p-value = 0.1439

• According to the results of the above permutation test, there is no association/correlation (p > 0.05) between wine excellence and residual sugar
• This p value of 0.1424 is higher than that of the Chi-Square test (.1386)

c. Can you conclude that “residual.sugar” is a significant factor contributing to the excellence of wine? Why?

• We cannot conclude that residual sugar is a significant factor contributing to the excellence of wine because the p value is not less than or equal to .05. To reject the null hypothesis that there is no association between wine excellence and residual sugar, the p value has to be less than or equal to .05