Difference in Prices in Different Supermarkets - Report Example

Case Analysis# Statistics Analysis Report for Decision making Student Name: XX Student No.: XX Words: 2951 Contents Contents i Executive summary ii Data for question 1 1 Data for Question 3 6 Solution for Question 4 8 Solution for Question 5 11 Executive summary The purpose of this study is to gauge the level of competitiveness in the Australian Supermarket prices for Basic Staple Basket. Hypothesis testing methodology was employed to carryout analysis. From the findings the following information was revealed. Firstly, ALDI supermarket is cheaper than other supermarkets with Woolworth and Cole average prices not statistically different from each other and similarly to the average prices of Independents and Cole/BI-LO. Secondly, there is no sufficient evidence to support the claim that there is a significance difference on average prices across the three states of New South Wales, Queensland and Victoria. Thirdly, the evidence is not sufficient to support the claim that there is a significant difference of average prices at different locations for Woolworth supermarkets. Finally, prices for ALDI stores in three states differences are not statistically significant and the price difference is due to State in which ALDI is operating. Data for question 1 Table 1.1: Descriptive statistics for Price of the basic staple basket among supermarkets Name of the supermarket Cole / BI-LO Woolworths/Safeways Independents ALDI Price of the basic staple basket Mean 83.296000 82.112333 84.473667 60.434667 95% Confidence Interval for Mean Lower Bound 81.247728 83.207405 60.204848 60.204848 Upper Bound 82.976938 85.739928 60.664485 60.664485 5% Trimmed Mean 83.153519 82.074815 84.412963 60.386481 Median 82.370000 81.520000 84.495000 60.250000 Variance 13.921 5.361 11.500 .379 Std. Deviation 3.7310270 2.3154544 3.3911106 .6154658 Minimum 77.7700 77.0000 77.6600 59.6100 Maximum 91.6500 88.1500 91.9500 62.6500 Range 13.8800 11.1500 14.2900 3.0400 Interquartile Range 3.7850 2.6400 4.4375 .8000 Skewness .826 .405 .292 1.511 Kurtosis -.099 1.188 -.169 4.373 From the data above, ALDI has the least mean price of the basic staple basket at ($ 60.43) with highly priced supermarket in terms of mean price being Independents ($ 84.47). However, it’s worth noting Independents supermarket has highest price as much as $ 91.95. The median, which is the value above and below, which half of all the cases fall within, in this case, the median for ALDI is $ 60.25 and that of Independents being $ 84.495. In regards to prices dispersion, ALDI has the least range of 3.04 with variance and standard deviation of .379 and .61547 respectively. On the other hand the prices of Independents supermarket have a wider range of 14.29 units with Cole/BI-LO having variance and standard deviation of 13.921 and 3.731. The prices in all supermarkets are positively skewed and closer to perfect normal distribution with ALDI recording highest frequencies in terms of items sold at particular price as denoted by kurtosis of 4.373. Figure 1.1: Boxplot for Price of the basic staple basket among supermarkets The box plot above shows boundaries of the box which are Turkey’s hinges representing the length of the box as the interquartile range based on Tukey’s hinges and are calculated as IQR=Q3−Q1. The prices of Independents supermarket have largest interquartile range of 14.2900 representing the spread of its middle 50% of prices, while ALDI with the least of 3.0400. The box plot also enables us to observe the sample price series if it may be contaminated by outliers. These are extreme values that may change the mean level of the uncontaminated series. The purpose of outlier detection is to find if there are outliers and what are their locations, their types, and magnitudes. In the above data there are no extreme outliers hence all prices are within range. Data for Question 2 Table 1.2: Descriptive statistics for Price of the basic staple basket among locations of the supermarkets As shown above the mean prices of Metropolitan ($ 77.62) and Others ($ 77.54) do not have a bigger difference margin as also depicted with its median of 81.68 and 81.195 respectively. However is worth noting ‘Other location’ of supermarket has higher variance and standard deviation, 109.507 and 10.4645357 respectively as compared to Metropolitan of 106.501 and 10.3199. The skewness of the prices are negatively and characterized by relatively lower frequencies for both locations as compared to normal distribution. Figure 1.2: Boxplot for Price of the basic staple basket among locations of the supermarkets The box plot above shows boundaries of the box which represent the length of the box as the interquartile range based on Tukey’s hinges and depicts negative skewness are higher frequencies of prices can be observed above median. Both locations don’t have extreme price values that may change the mean level of the uncontaminated series statistics calculated. Data for Question 3 Table 1.3: Descriptive statistics for the prices of basic staple basket of ALDI stores ALDI Statistic Std. Error Price of the basic staple basket Selected Mean 60.434667 .1123682 95% Confidence Interval for Mean Lower Bound 60.204848 Upper Bound 60.664485 5% Trimmed Mean 60.386481 Median 60.250000 Variance .379 Std. Deviation .6154658 Minimum 59.610 Maximum 62.6500 Range 3.0400 Interquartile Range .8000 Skewness 1.511 .427 Kurtosis 4.373 .833 The above results represent descriptive statistics for the prices of basic staple basket of ALDI stores. The mean price is $ 60.434667 which was the lowest among the supermarkets. It has a median of 60.25, variance of .379 and standard deviation which is a square root of variance of .6154658. The prices range is 3.04 with its middle 50% prices ranging within a margin of .800 depicting a closely priced item in its stores. Figure 1.3: Box plot for the prices of basic staple basket of ALDI stores The box plot above shows boundaries of the box which are positively skewed (1.511). The prices of ALDI supermarket have smallest interquartile range representing the spread of its middle 50% of prices. The box plot also enables us to observe the sample price series if it may be contaminated by outliers in which ALDI contain only one item priced at $ 62.65 and no extremely priced items to warrant meaningful changes to the mean level of the uncontaminated series. Solution for Question 4 Assume that, ,  and  are the population means prices of Independent, ALDI, Coles and Woolworth supermarkets. In this case, we want to test the claim that there is no significance difference of the average prices for the basic staple basket at different supermarket. Step 1: Set up H0 and Ha The test is two-tailed because we want to examine deviations from the null in two directions. Then the null and alternative hypotheses are: H0:  =  =  =  Ha: At least one of the population means is different from others. Step 2: Decide on the type of test The appropriate test is F test calculated from one-way ANOVA as there is only one dependent variable. Step 3: Decide on level of significance and determine the critical value and region The significance level in this case has been given at  with df = n1 + n2 +n3 +n4 – 4 = 30 + 30 + 30 + 30 – 4 = 116. The critical values are shown in table xx below. Step 4: Write down the decision rule If the computed p- value using the ANOVA output is less than α value of 0.05, the null hypothesis should be rejected. Otherwise, we fail to reject the null hypothesis. Step 5: Select a random sample and do relevant calculations From the data provided, the following values can be obtained SPSS. Table 1.4: Descriptive Price of the basic staple basket N Mean Std. Dev. Std. Error 95% Confidence Interval for Mean Min Max Lower Bound Upper Bound Cole/BI-LO 30 83.2960 3.73103 .681189 81.902812 84.689188 77.7700 91.6500 Woolworths/Safeways 30 82.1123 2.31545 .422742 81.247728 82.976938 77.0000 88.1500 Independents 30 84.4737 3.39111 .619129 83.207405 85.739928 77.6600 91.9500 ALDI 30 60.4347 .61547 .112368 60.204848 60.664485 59.6100 62.6500 Total 120 77.5792 10.3488 .944712 75.708541 79.449792 59.6100 91.9500 Table 1.4: ANOVA Price of the basic staple basket Sum of Squares df Mean Square F Sig. Between Groups 11840.994 3 3946.998 506.670 .000 Within Groups 903.649 116 7.790 Total 12744.643 119 Table 1.6: Post Hoc Tests Multiple Comparisons Dependent Variable: Price of the basic staple basket (I) Name of the supermarket (J) Name of the supermarket Mean Difference (I-J) Std. Error Sig. 95% Confidence Interval Lower Bound Upper Bound Games-Howell Cole/BI-LO Woolworths/Safeways 1.1836667 .8017043 .459 -.949258 3.316591 Independents -1.1776667 .9205106 .580 -3.613167 1.257834 ALDI 22.8613333* .6903951 .000 20.986109 24.736557 Woolworths/Safeways Cole/BI-LO -1.1836667 .8017043 .459 -3.316591 .949258 Independents -2.3613333* .7496879 .014 -4.352084 -.370583 ALDI 21.6776667* .4374215 .000 20.494609 22.860724 Independents Cole/BI-LO 1.1776667 .9205106 .580 -1.257834 3.613167 Woolworths/Safeways 2.3613333* .7496879 .014 .370583 4.352084 ALDI 24.0390000* .6292437 .000 22.330906 25.747094 ALDI Cole/BI-LO -22.8613333* .6903951 .000 -24.736557 -20.986109 Woolworths/Safeways -21.6776667* .4374215 .000 -22.860724 -20.494609 Independents -24.0390000* .6292437 .000 -25.747094 -22.330906 *. The mean difference is significant at the 0.05 level. Table 1.7: Price of the basic staple basket Name of the supermarket N Subset for alpha = 0.05 1 2 3 Tukey Ba ALDI 30 60.434667 Woolworths/Safeways 30 82.112333 Cole/BI-LO 30 83.296000 83.296000 Independents 30 84.473667 Means for groups in homogeneous subsets are displayed. a. Uses Harmonic Mean Sample Size = 30.000. Step 6: Draw a conclusion The decision; reject the null hypothesis as the mean price differences are statistically significant for . The post hoc test provides further source of the observed differences as average prices of ALDI supermarket being different and cheaper than other supermarkets. Woolworth and Cole average prices are not statistically different from each other and similarly to average prices of Independents and Cole/BI-LO. Solution for Question 5 Assuming that µ1, µ2 and µ3 are the population mean prices of the Basic Staple Basket for New South Wales, Queensland and Victoria States respectively. Step 1: Set up H0 and Ha H0: µ1= µ2= µ3 Ha: At least one of the population means is different from the others. Step 2: Decide on the type of test As there is only one dependent variable, the appropriate test is the F test calculated from one-way ANOVA ，The F ratio for testing equality of group means is; F= Mean Square Between/ Mean Square Within =BSSM/WSSM Step 3: Decide on the level of significance and determine the critical value and region. The value of α is 0.05. The critical value is shown in the following Table xx. Step 4: Write down the decision rule If the computed p- value using the ANOVA output is less than α value of 0.05, the null hypothesis should be rejected. Otherwise, we fail to reject the null hypothesis. Step 5: Select a random sample and do relevant calculations From the data provided, the following values can be obtained using SPSS. Table 1.8: Descriptive Price of the basic staple basket N Mean Std. Deviation Std. Error 95% Confidence Interval for Mean Minimum Maximum Lower Bound Upper Bound New South Wales 40 77.013500 9.9285986 1.5698493 73.838180 80.188820 59.6900 86.5100 Queensland’s 40 77.640750 10.8461524 1.7149273 74.171982 81.109518 59.6100 91.9500 Victoria 40 78.083250 10.4869891 1.6581386 74.729348 81.437152 60.7500 90.4100 Total 120 77.579167 10.3488087 .9447127 75.708541 79.449792 59.6100 91.9500 Table 1.9: ANOVA Price of the basic staple basket Sum of Squares df Mean Square F Sig. Between Groups 23.115 2 11.557 .106 .899 Within Groups 12721.528 117 108.731 Total 12744.643 119 Step 6: Draw a conclusion The decision fail to reject the null hypothesis as the p-value .899 is greater than α value of 0.05. The evidence is not sufficient to support the claim that there is a significance difference on average prices across the three states of New South Wales, Queensland and Victoria. Read More