Question: Statistics - Case Study 1 Contents Executive Summary...................................................................................................................................3 1. Business Problem...............................................................................................................................4 2. Analysis Tools....................................................................................................................................4 3. Data Analysis......................................................................................................................................4 3.1. Part 1..........................................................................................................................................4 Gender....................................................................................................................................................4 Fragrance Type.......................................................................................................................................7 Country of
Statistics - Case Study 1 Contents Executive Summary...................................................................................................................................3 1. Business Problem...............................................................................................................................4 2. Analysis Tools....................................................................................................................................4 3. Data Analysis......................................................................................................................................4 3.1. Part 1..........................................................................................................................................4 Gender....................................................................................................................................................4 Fragrance Type.......................................................................................................................................7 Country of Production............................................................................................................................8 3.2. Part 2........................................................................................................................................13 3.2.1. 3.3. Descriptive Analysis........................................................................................................13 Part 3........................................................................................................................................18 3.3.1. Determine if average prices for female products are greater than $140......................18 3.3.2. Determine if average prices for female products exceed average prices for male products 19 3.3.3. Determine if average prices for perfume exceeds the average price of cologne..........20 3.3.4. Determine if there is a difference in the prices for intensity.........................................22 3.3.5. Determine if there is a difference for the prices for different countries.......................23 4. Overall Conclusion............................................................................................................................24 5. Implications and recommendations................................................................................................24 Statistics - Case Study 2 Executive Summary In this case study analysis statistical analyses have been performed and descriptive analysis, t-test, F-test and analysis of variance have been carried out. As per the findings from analysis, it can be concluded that it is viable for the company to manufacture perfumes for females which are strong. Statistics - Case Study 3 Case Study - Australian Manufacturing Company 1. Business Problem An Australian business which is a market leader in sanitary product sector is considering expanding the business to Australian and Asian markets with an addition to its product line. The business has decided to introduce fragrance products for which Research and Development Department of the company has been provided with the data of 95 different fragrance products. A statistical analysis is done for better understanding of the markets, and for an appropriate proposition of price. 2. Analysis Tools Data is analysed using MS Excel on which various tests i.e. t, z and fhave been performed and different hypotheses have been tested. The above mentioned tests have helped in understanding the markets and interpretations have been made on the results by a qualified analyst. 3. Data Analysis 3.1. Part 1 Description of Data: The data contains 95 different fragrances which are segregated in columns of Gender, Fragrance Type, Originating Country and Intensity. Along with these divisions the respective price of each product is also provided. For better understanding the analyst has used charts to explain. Gender Statistics - Case Study 4 Gender 29% Female Male 71% Figure 1 Figure 1 explains the gender of the fragrances chosen. It is clear that out of those chosen 95 fragrances 29% are male fragrances and 71% are female fragrances. Male 29 29 36 57 65 66 69 69 69 74 97 111 116 117 119 123 133 135 155 157 159 Female 34 36 38 39 41 43 43 43 45 46 48 49 51 51 51 55 55 55 65 65 65 Statistics - Case Study 5 175 177 183 183 185 185 193 66 69 69 73 75 83 89 104 112 119 124 126 133 139 139 143 143 159 163 163 173 183 186 188 199 207 207 207 213 217 223 243 247 247 247 252 257 265 267 267 277 277 279 287 305 307 Table 1 Table 1 show the respective prices of 28 male and 67 female fragrances. The minimum price of a male fragrance is 29 whereas the maximum price for a male fragrance is 193. For female fragrances the minimum price is 34 and the most expensive female fragrance chosen is 307. Statistics - Case Study 6 Note: All prices are in dollars. Fragrance Type Fragrance Type 54% 46% Perfume Colonge Figure 2 Figure 2 shows percentage of different fragrance types. From the above figure it is clear that 46% are perfumes and 54 % are colognes out of the selected 95 products. Cologne 29 34 36 36 38 39 41 43 43 43 45 46 48 49 Perfume 29 69 69 73 74 83 111 112 117 119 133 139 143 143 Statistics - Case Study 7 51 51 51 55 55 55 57 65 65 65 65 66 66 69 69 69 75 89 97 104 116 119 123 124 126 133 135 139 155 157 159 163 175 183 183 185 193 159 163 173 177 183 185 186 188 199 207 207 207 213 217 223 243 247 247 247 252 257 265 267 267 277 277 279 287 305 307 Table 2 Table 2 show the respective prices of 51 colognes and 44 perfumes. The minimum price of cologne is 29 whereas the maximum price for cologne is 193. For perfumes the minimum price is 29 and the most expensive cologne chosen is 307. Note: All prices are in dollars. Country of Production Statistics - Case Study 8 Country of production 26% 27% France USA OC 46% Figure 3 Figure 3 illuminates the countries where the fragrances are made. Out of the chosen 95 fragrances 26% are made in France, 46% are made in USA and the rest 28% are made in other countries. France 29 29 43 51 51 65 69 69 104 112 123 124 133 139 143 163 186 217 247 247 257 267 267 277 OC 34 36 38 39 46 48 66 66 69 74 119 119 126 143 155 163 173 175 183 183 193 199 213 223 USA 36 41 43 43 45 49 51 55 55 55 57 65 65 65 69 69 73 75 83 89 97 111 116 117 Statistics - Case Study 9 279 243 252 133 135 139 157 159 159 177 183 185 185 188 207 207 207 247 265 277 287 305 307 Table 3 Table 3 represent the respective prices of the products that are made in France, USA and Other Countries. It can be clearly seen the cheapest fragrance chosen from France is 29, USA is 36 and from the other countries is 34, whereas the most expensive fragrance chosen from France is 279, USA is 252 and form other countries is 307. Note: All prices are in dollars. Statistics - Case Study 10 Intensity Intensity 35% 36% Strong Medium Mild 29% Figure 4 Figure explains the percentage of different levels of intensity of fragrances chosen. It is shown from the above pie chart that out of the chosen 95 fragrances 35% are Strong, 29% are medium whereas, 36% are mild. Medium 39 57 69 69 73 83 89 97 111 112 117 119 119 123 124 133 133 135 Mild 29 29 34 36 36 38 41 43 43 43 45 46 48 49 51 51 51 55 Strong 143 155 163 173 175 183 183 183 185 185 193 199 207 207 207 213 217 223 Statistics - Case Study 11 139 139 143 157 159 159 163 177 186 188 55 55 65 65 65 65 66 66 69 69 69 74 75 104 116 126 243 247 247 247 252 257 265 267 267 277 277 279 287 305 307 Table 4 Table 4 represent the respective prices of the products according to their intensity level. It is visible that the least expensive medium intensity fragrance is priced at 39, and the most expensive medium fragrance chosen is for 188. From the second column it can be seen that the least expensive mild fragrance is for 29 whereas the most expensive is priced at 126. In column 3 the most expensive strong fragrance is 307 and the least is for 143. Note: All prices are in dollars. Mean Mean of the above data would represent the average price of the fragrances chosen according to their Gender, Type, Originating Country and Intensity. The means would be different if calculated keeping different conditions in place. Median Median is normally the value that divides the data into two halves. It is calculated by arranging the data into ascending order and then finding the middle value. Extreme values don't affect median. Keeping this data in the view the median would divide the prices into two halves under each division. Statistics - Case Study 12 Mode The value that is repeated the most in a data is mode. The mode will be the price that most of the fragrances will have out of the chosen 95 fragrances. Standard Deviation The amount of variation by which the values in a data is far from the mean is called standard deviation. Standard deviation in this scenario will be the deviation in prices of products from the average calculated price. Coefficient of Variation The standardized dispersion of the prices of different fragrances from each other under different divisions such as price according to gender, price according to intensity etc would be the coefficient of variation of the prices. 3.2. Part 2 In this section of the report, analyses of the prices are done with four different categories mentioned above. 3.2.1. Descriptive Analysis Various estimates including mean, median, mode, standard deviation and coefficient of variation have been calculated and interpreted in this part using Excel. These analyses have been performed keeping confidence level at 95% which means that the analyst is 95% confident that the calculated range would contain the true population mean. Analyses have been performed on the following groups. Prices for gender - male and female. Prices for fragrance - perfume and cologne. Prices for intensity - strong, medium and mild. Prices for different countries - France, USA and other. Statistics - Case Study 13 Prices for Gender - Male and Female The descriptive statistics for prices for gender based classification are presented as follows: Male Mean Median Mode Standard Deviation Coefficient of Variation Confidence Level(95.0%) Female 116.6428571 118 69 53.1512268 46% 20.60989395 Mean Median Mode Standard Deviation Coefficient of Variation Confidence Level(95.0%) 142.3283582 133 43 87.63999849 62% 21.37707507 The analysis shows that mean and median prices of fragrances of females are more than the males; however the mode of male fragrances i.e. 69 is higher than the females' 43, which means that prices for male fragrances are more repetitive than female fragrances prices. Standard Deviation from the mean of female fragrances is higher than male fragrances showing that there is a higher price difference amongst the female fragrances than the price difference of male fragrances. Moreover, the standardized measure of dispersion of female fragrances prices is more than the male fragrances making the prices more dispersed. Furthermore, the probability of population mean falling within the range of female fragrances prices is more than the chances of population mean falling in male fragrances prices, as evidenced from the value of 95% confidence interval. Finally, male fragrances prices distribution is negatively skewed as the mean of the data is less than the median. For female fragrances prices distribution the mean of the data is more than the median hence it is positively skewed. Prices for Fragrance - Perfume and Cologne The descriptive statistics for prices for fragrance based classification are presented as follows: Cologne Mean Median Mode Standard Deviation Coefficient of 87.78431373 66 65 49.64768423 57% Perfume Mean Median Mode Standard Deviation Coefficient of Variation 189.2045455 193.5 207 73.13942707 39% Statistics - Case Study 14 Variation Confidence 13.96363593 Confidence Level(95.0%) 22.23641797 Level(95.0%) The analysis shows that mean, median and mode of prices of perfumes are more than the prices of colognes. This means that the average prices of perfumes are more than the prices of colognes in the market. Standard Deviation from the mean of perfumes is higher than colognes showing that there is a higher price difference amongst the perfumes than the price difference of colognes. Moreover, the standardized measure of dispersion of cologne prices is more than the perfume prices making the cologne prices more wide spread. Furthermore, the probability of population mean falling within the range of perfume average prices is more than the chances of population mean falling in cologne prices, as evidenced from the value of 95% confidence interval. Finally, perfume prices distribution is negatively skewed as the mean of the data is less than the median. For cologne prices distribution the mean of the data is more than the median hence it is positively skewed. Prices for different countries - France, USA and other The descriptive statistics for prices for country of manufacture based classification France Mean Median Mode Standard Deviation Coefficient of Variation Confidence OC 147.64 133 29 87.16214775 Mean Median Mode Standard Deviation 59% Coefficient of Variation 35.97876602 Confidence USA 129.9230769 134.5 66 71.32456692 Mean Median Mode Standard Deviation 55% Coefficient of Variation 28.8086158 Confidence 130.2954545 113.5 55 80.85145137 62% 24.58108763 Statistics - Case Study 15 Level(95.0%) Level(95.0%) Level(95.0%) The analysis showed that the average price of fragrances made in France is more than the average prices of USA and Other countries. Meaning that the fragrances made in France are expensive than USA and other countries. Moreover, median of other countries is the highest amongst the other two comparisons. In addition the mode of other countries is also the highest, which means that other countries' fragrance prices are more repetitive than USA and France. The Standard Deviation of French made fragrances is the highest and most deviated from the mean. The standardized measure of dispersion of USA made fragrances prices are more than any of the above mentioned countries. This means that the prices of fragrances made in USA are more wide spread. The confidence level 95% shows that the French made fragrances represent the actual population mean most appropriately. Finally, France made fragrances distribution is positively skewed as the mean is greater than the median along with the USA made products prices, because the mean of this distribution is also greater than its median. Only prices distribution of other countries' fragrances is negatively skewed as mean is less than the median. Prices for Intensity - Strong, Medium and Mild The descriptive statistics for prices for intensity based classification are presented as follows: Medium Mean Median Mode Standard Deviation Coefficient of Variation Confidence Mild 121.8571429 Mean 123.5 Median 69 Mode 39.0571417 Standard Deviation 32% Coefficient of Variation 15.14477834 Confidence Strong 58 Mean 53 Median 65 Mode 22.42428337 Standard Deviation 39% Coefficient of Variation 7.824202318 Confidence 224.787878 8 217 183 45.4132122 7 20% 16.1028248 Statistics - Case Study 16 Level(95.0%) Level(95.0%) Level(95.0%) The analysis clearly states that the average prices of strong fragrances are higher than medium and mild fragrances. Median and Mode of the strong fragrances are also the highest. It can be said that overall the prices of strong intensity fragrances are more than medium and mild intensity fragrances. Moreover, a greater standard deviation is witnessed in strong intensity fragrances' prices than the mild and medium intensity prices. The coefficient of variation for mild is the higher than both the chosen intensities. The confidence interval of 95% represents that the strong intensity fragrances presents the population means most appropriately. Lastly the distribution for mild intensity fragrances is negatively skewed because of lower mean; whereas, the mild and strong intensity fragrances distribution is positively skewed because the mean of both the distributions are higher than their respective medians. 3.3. Part 3 In this part, data analysis has been performed and the same has been interpreted for six individual cases as follows: 3.3.1. Determine if average prices for female products are greater than $140. Assumption The mean or average prices for female products = f Following are the null and alternate hypotheses: H0: Average prices for female products are less than equal to $140( f $140/ml ) Ha: Average prices for female products are greater than $140( f > $140/ml ) Choice of Test Since population standard deviation is unknown, therefore, t-test is appropriate choice for analysis, which is conducted using the following formula: 1 Statistics - Case Study 17 Sample size for female products is 67, the degree of freedom therefore is 66 which is calculated by D.F. =n-1, =.05, t.05, 66 = 1.668. Decision Rule In this one tailed test the null hypothesis will be rejected if calc t. is greater than critical t. Calculation Putting the values into the formula we get t= 140142.30 87.64 /8.19 t = -0.21 Test for Validity of Null Hypothesis Since t value calculated is less than the critical value, therefore the null hypothesis shall not be rejected. Interpretation From the analysis it can be said that there is not enough evidence to suggest that average prices for female products are greater than $140 per 100ml. 3.3.2. Determine if average prices for female products exceed average prices for male products Assumptions The mean or average prices for female products = f The mean or average prices for male products = m Following are the null and alternate hypotheses: H0: f - m 0 Ha: f - m > 0 Statistics - Case Study 18 Choice of Test Standard Deviation is unknown, t test will be used. t= ( x 1 x 2 ) ( 12 ) 2 2 s 1 ( n11 ) + s2 ( n21 ) 1 1 + n 1+ n22 n1 n2 The sample size of male products' prices is 28 whereas for female products' prices the sample size is 67. Therefore degrees of freedom will be n 1+n2-2. Therefore is 93, =.05, t.05, 93 = 1.661. Decision Rule The null hypothesis will be rejected if calc t. is greater than critical t. The value for critical t is 1.661. Calculation Putting values in the formula to get calc. t t= ( 142.33116.64 ) ( 0 ) 87.642 ( 671 )+53.15 2 ( 281 ) 1 1 + 67+ 282 67 28 t = 1.443 Test for Validity of Null Hypothesis 1.443<1.661 which means that null hypothesis shall not be rejected. Interpretation From the analysis it is eminent that there is not enough evidence available to support the claim that the average prices of female products is greater than the average Statistics - Case Study 19 prices of male products; which means that there will be no significant difference in the profits obtained or will be obtained by female or male products. 3.3.3. Determine if average prices for perfume exceeds the average price of cologne Assumptions The mean or average prices for perfumes products = p The mean or average prices for cologne products = c Following are the null and alternate hypotheses: H0: p - c 0 Ha: p - c > 0 Choice of Test Since population standard deviation is unknown, therefore, t-test is appropriate choice for analysis, which is conducted using the following formula: t= ( x 1 x 2 ) ( 12 ) s 2 ( n11 ) + s2 ( n21 ) 1 1 1 2 + n 1+ n22 n1 n2 Keeping in view the fact that the size of sample for perfume products' prices is 44 and for cologne products' prices is 51, the degree of freedom therefore is 93, =.05, t.05, 93 = 1.661. Decision Rule Considering the fact that the t-test is one-tailed, therefore if the t value observed is higher than 1.661, then the null hypothesis shall be rejected. Calculation Statistics - Case Study 20 Replacing the values in the formula noted earlier, following t value is obtained: t= ( 189.2087.78 )( 0 ) 73.142 ( 441 ) +49.652 ( 511 ) 1 1 + 44+512 51 44 t = 7.99 Test for Validity of Null Hypothesis Since t value calculated is higher than the critical value, therefore the null hypothesis shall be rejected. Interpretation From the analysis it can be said that there is enough evidence to suggest that average prices for perfume products are greater than average price for cologne products. So it can be helpful for profits of the company if perfumes are made more than colognes. 3.3.4. Determine if there is a difference in the prices for intensity Choice of Test F test is performed in order to perform this analysis moreover single factor ANOVA has also been used. Following are the null and alternate hypotheses: H0: 1=2= 3 Ha: At least one of the three means is different Decision Rule Statistics - Case Study 21 If the calc. F value turns out to be higher than the critical value from F distribution, the null hypothesis shall be rejected. From the F table the critical value is F=F.05, 2,92 =3.07 If Ftest>Fcriti, Reject H0; If Ftest<=Fcriti, Do not Reject H0; Calculation Anova: Single Factor SUMMARY Groups Count Medium 28 Mild 34 Strong 33 ANOVA Source of Variation Between Groups Within Groups Total SS 472458.487 9 123776.943 7 596235.431 6 Su m 341 2 197 2 741 8 Averag e 121.857 1 58 224.787 9 df 2 92 Varianc e 1525.46 502.848 5 2062.36 MS F 236229. 2 1345.40 2 175.582 7 Pvalue 3.91E32 F crit 3.09543 3 94 Test for Validity of Null Hypothesis Calculations above represents clearly that Ftest>Fcriti therefore we will reject H0. Interpretation The test shows that all the means of strong, mild and medium fragrances are not the same; at least one of them is different. There is a significant difference between the prices of different intensities of fragrances. 3.3.5. Determine if there is a difference for the prices for different countries Statistics - Case Study 22 Choice of Test F test is performed in order to perform this analysis moreover single factor ANOVA has also been used. Following are the null and alternate hypotheses: H0: 1=2= 3 Ha: At least one of the three means is different Decision Rule If the calc. F value turns out to be higher than the critical value from F distribution, the null hypothesis shall be rejected. From the F table the critical value is F=F.05, 2,92 =3.07 If Ftest>Fcriti, Reject H0; If Ftest<=Fcriti, Do not Reject H0; Calculation Anova: Single Factor SUMMARY Groups Count Sum Average 25 26 3691 3378 USA 44 5733 ANOVA Source of Variation Between Groups Within Groups Total SS 5632.66633 4 590602.765 2 596235.431 6 df 2 92 147.64 129.923 1 130.295 5 MS France Other Countries Varianc e 7597.24 5087.19 4 6536.95 7 F P-value F crit 2816.33 3 6419.59 5 0.43870 9 0.6462 1 3.09543 3 94 Test for Validity of Null Hypothesis The calculation shows that Ftest
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