100742_Homework-9-Assignment

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Jan 9, 2024

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Homework Assignment-9 Problem-1 H 0: μ 1= μ 2= μ 3 H 1: At least one  μi  is different Anova: Single Factor SUMMARY Groups Count Sum Average Variance Store 1 5 225 45 3.5 Store 2 4 144 36 4.666667 Store 3 3 99 33 4 ANOVA Source of Variation SS df MS F P-value F crit Between Groups 324 2 162 40.5 3.16E- 05 4.256495 Within Groups 36 9 4 Total 360 11
Because, p=3.16E-05 is < 0.05, reject H0. This suggests that there is a significant difference in the average sales of the three stores. Problem-2 H0: The mean sales for all box colors are equal. H1: At least one box color has a different mean sale. Anova: Single Factor SUMMARY Groups Count Sum Average Variance Red 6 372 62 205.6 Yellow 5 265 53 261 Blue 5 260 52 384 ANOVA Source of Variation SS df MS F P-value F crit Between Groups 340.9375 2 170.4688 0.614217 0.556046 3.805565 Within Groups 3608 13 277.5385 Total 3948.938 15
Because, p=.5560 is > 0.05, fail to reject H0. This suggests that the mean sales for all box colors are equal. Problem-3 H0: The mean tune-up time is the same for both analyzers. H1: There is a significant difference in the mean tune-up time between the two analyzers. Anova: Two-Factor Without Replication SUMMARY Coun t Su m Averag e Varianc e Compact 2 92 46 32 Intermediate 2 99 49.5 60.5 Full-sized 2 10 9 54.5 144.5 Computerized 3 16 8 56 43 Electronic 3 13 2 44 4 ANOVA Source of Variation SS df MS F P-value F crit Rows 73 2 36.5 3.47619 0.22340 19
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4 Columns 216 1 216 20.5714 3 0.04533 1 18.5128 2 Error 21 2 10.5 Total 310 5 The F for testing whether the time depends on the type of engine analyser is 20.57143. p-value is 0.0453, which is less than 0.05. The null hypothesis is rejected. Therefore, there is a significant difference in the mean tune-up time between the two analyzers. Problem-4 H0: μ = μ = μ = μ (There is no significant difference in the means) ₁ ₂ ₃ ₄ H1: At least one mean is different from the others (There is a significant difference in the means) Anova: Two-Factor Without Replication SUMMARY Count Su m Average Variance i 4 335 83.75 22.91667 4 335 83.75 16.91667 ii 4 357 89.25 30.91667
4 355 88.75 64.91667 iii 4 317 79.25 10.91667 4 309 77.25 24.25 a 6 499 83.16667 70.16667 b 6 519 86.5 43.5 c 6 481 80.16667 7.366667 d 6 509 84.83333 49.36667 ANOVA Source of Variation SS df MS F P-value F crit Rows 470.8333 5 94.16667 3.705728 0.022022 2.901295 Columns 131.3333 3 43.77778 1.722781 0.205146 3.287382 Error 381.1667 15 25.41111 Total 983.3333 23 The F for testing the difference in yield depends on the type of fertilizer is 1.722781. p-value is 0.205146, which is > than 0.05. The null hypothesis is not rejected. There is no significant difference in the means. Thus, the yields do not depend on the type of fertilizer. Case Study: TourisTopia Travel Descriptive Statistics
White background Ariel Calibri Tahoma Mean 310.4 252.5 251.5 Median 299.5 251.5 252 Standard Deviation 41.9741 49.0334 188.4164 Sample Variance 1761.822 2404.278 339.167 Green background Ariel Calibri Tahoma Mean 252.9 241 249.2 Median 261.5 232 257.5 Standard Deviation 49.38837 55.71555 43.48129 Sample Variance 2439.211 3104.222 1890.622 Pink background Ariel Calibri Tahoma Mean 253.2 234.9 206.4 Median 260 251.5 202.5 Standard Deviation 49.06413 61.73321 28.61701 Sample Variance 2407.289 3810.989 818.9333 Mean and median times spent seem to vary across background colors. Mean and median times spent vary across font types. Standard deviations indicate variability within each group.
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Preliminary observations suggest variations in mean and median times across differen0t combinations. Further statistical analysis (ANOVA) is needed to confirm if these differences are significant. 2. Observational Study or Controlled Experiment Triple T has used a controlled experiment. In a controlled experiment, the researcher manipulates one or more independent variables (in this case, background color and font type) to observe the effect on a dependent variable (in this case, time spent by visitors). In this study, Triple T's IT group designed prototype homepages featuring different combinations of background colors and fonts and implemented computer code to randomly direct each website visitor to one of these prototype homepages. This random assignment is a key characteristic of controlled experiments, allowing for the comparison of different treatments and reducing the likelihood of confounding variables. 3. Hypothesis Testing: Background Color Null Hypothesis (H0): Pink μ White= μ Green= μ Pink (Time spent is equal for all background colors) α =0.05. ANOVA Source of Variation SS df MS F P-value F crit Rows 98988.22 29 3413.387 1.829236 0.025424 1.662901 Columns 22426.29 2 11213.14 6.00913 0.004249 3.155932 Error 108229 58 1866.018 Total 229643.6 89
based on the ANOVA results, there is evidence to suggest that the time spent by visitors to the Triple T website differs across the three background colors (white, green, and pink) becaue p=.0254<0.05. 4. Hypothesis Testing: Font Null Hypothesis (H0):Tahoma μ Arial= μ Calibri= μ Tahoma (Time spent is equal for all fonts) α =0.05. Anova: Single Factor ANOVA Source of Variation SS df MS F P-value F crit Between Groups 22426.28889 2 11213.14444 4.70783 0.011446 3.101296 Within Groups 207217.2667 87 2381.807663 Total 229643.5556 89 Based on the ANOVA results for the fonts (Arial, Calibri, Tahoma), there is evidence to suggest that the time spent by visitors to the Triple T website differs across the three fonts because p=0.0114<0.05. 5. Hypothesis Testing: Combination of Background Color and Font Null Hypothesis (H0): All means for combinations are equalAll means for combinations are equal
Anova: Two-Factor With Replication SUMMARY Ariel Calibri Tahoma Total white background Count 10 10 10 30 Sum 3104 2525 2515 8144 Average 310.4 252.5 251.5 271.4667 Variance 1761.822222 2404.278 339.1666667 2182.395 green background Count 10 10 10 30 Sum 2529 2410 2492 7431 Average 252.9 241 249.2 247.7 Variance 2439.211111 3104.222 1890.622222 2332.7 pink background Count 10 10 10 30 Sum 2532 2349 2064 6945 Average 253.2 234.9 206.4 231.5 Variance 2407.288889 3810.989 818.9333333 2567.569 Total Count 30 30 30 Sum 8165 7284 7071 Average 272.1666667 242.8 235.7 Variance 2806.971264 2947.338 1391.113793 ANOVA Source of Variation SS df MS F P-value F crit Sample 24246.28889 2 12123.14444 5.749643 0.004624 3.109311 Columns 22426.28889 2 11213.14444 5.318058 0.00676 3.109311 Interaction 12182.17778 4 3045.544444 1.44441 0.226942 2.484441 Within 170788.8 81 2108.503704 Total 229643.5556 89
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There is no compelling evidence, according to the ANOVA results for the combination of background color and font, that the amount of time visitors spend on the Triple T website varies depending on the combination. p=0.2269>0.05 indicates that the interaction effect is not statistically significant. 6. Results and Recommendations Background Color: The backdrop color factor's p-value is 0.025424, which is below the 0.05 significance level (α). As a result, the null hypothesis would be rejected. This implies that there is sufficient data to draw the conclusion that users of the Triple T website spend varied amounts of time on different backdrop colors. Font: The font type factor's p-value is 0.011446, which is below the 0.05 significance level (α). As a result, the null hypothesis would be rejected. This implies that there is sufficient data to draw the conclusion that users of the Triple T website spend varied amounts of time with different fonts. Combination of Background Color and Font: The interaction term (background color and font combination) has a p-value of 0.226942, above the significance level (α) of 0.05. You wouldn't reject the null hypothesis as a result. This implies that there is insufficient data to draw the conclusion that users of the Triple T website spend different amounts of time on different backdrop color and font combinations. Recommendation:
Based on the statistical significance of the background color and font separately, it appears that these elements significantly affect how long visitors stay on the page. The interaction effect, which is the result of the font and background color combination, does not, however, appear to have much of an effect. As a result, in order to improve user experience and possibly lengthen visitors' stay on the website, Triple T should think about optimizing the background color and typeface separately. The best mix may be discovered through additional research or A/B testing.

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