Introduction
This project is the statistic project on Randomize design, ANOVA and paired t test. For this project I have collected Breakfast Product price as a data from the three stores. They are Family Fare, Hornbachers and Wallmart.
Completely Randomize Design and Dunnett’s Comparison:
In this part we have analyzed a data using completely randomize design. This is use to test whether the mean prices of product I have selected from three stores is same of difference. Besides this I have used Dunnett’s comparison using Family Fare as the base store for comparison. This compare the mean prices of other two stores with the price of Family Fare. Output from the Minitab is shown in section Result from Minitab and I will discuss about the result in section Finding and Discussion.
Result from Minitab:
One-way ANOVA: Family fare, Hornbachers, Wallmart
Method
| Null hypothesis | All means are equal |
| Alternative hypothesis | Not all means are equal |
| Significance level | α = 0.05 |
Equal variances were assumed for the analysis.
Factor Information
| Factor | Levels | Values |
| Factor | 3 | Family fare, Hornbachers, Wallmart |
Analysis of Variance
| Source | DF | Adj SS | Adj MS | F-Value | P-Value |
| Factor | 2 | 12.34 | 6.171 | 2.84 | 0.066 |
| Error | 63 | 136.81 | 2.172 | ||
| Total | 65 | 149.15 |
Model Summary
| S | R-sq | R-sq(adj) | R-sq(pred) |
| 1.47364 | 8.27% | 5.36% | 0.00% |
Means
| Factor | N | Mean | StDev | 95% CI |
| Family fare | 22 | 4.381 | 1.418 | (3.753, 5.009) |
| Hornbachers | 22 | 4.447 | 1.730 | (3.819, 5.075) |
| Wallmart | 22 | 3.498 | 1.230 | (2.871, 4.126) |
Pooled StDev = 1.47364
Dunnett Multiple Comparisons with a Control
Grouping Information Using the Dunnett Method and 95% Confidence
| Factor | N | Mean | Grouping |
| Family fare (control) | 22 | 4.381 | A |
| Hornbachers | 22 | 4.447 | A |
| Wallmart | 22 | 3.498 | A |
Means not labeled with the letter A are significantly different from the control level mean.
Dunnett Simultaneous 95% CIs
Interval Plot of Family fare, Hornbachers, …
Finding and Discussion:
In this part we have used breakfast mean price data from all three stores. We have set Null Hypothesis as all mean are equal vs Alternate Hypothesis as all mean price are not equal. Significance level value alpha is set to 0.05. Then from Minitab we get F-value equal to 2.84 and p-value 0.066. Which means p-value is greater than that of alpha which means “Don’t reject Null Hypothesis”. Which also tells that the differences between the means are not statistically significant.
Furthermore, we have used Dunnett’s comparison in Minitab. For this we have used Family fare as the base store for the comparison and compare the mean price of Hornbachers and Walmart. We have got mean price value of 4.381, 4.447 and 3.498 for Family Fare, Hornbachers and Walmart respectively. Both the Family fare – Hornbachers and Family Fare – Walmart interval contain zero which means corresponding mean(Hornbachers and Walmart mean) is not significantly different from the control mean(Family Fare). Last figure shows the interval plot for all three stores.
Stat-ANOVA-Balanced ANOVA:
The second type of design is randomized complete block design. This test whether the mean prices are the same between the stores. Result from Minitab and findings is in section below. For this experiment Store Number 1 is for Family Fare, 2 is for Hornbachers and 3 is for Walmart. Similarly, digit 1 to 22 item number is given to 22 different breakfast items.
Result from Minitab:
ANOVA: Price versus Store Number, Item Number
Factor Information
| Factor | Type | Levels | Values |
| Store Number | Fixed | 3 | 1, 2, 3 |
| Item Number | Fixed | 22 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 |
Analysis of Variance for Price
| Source | DF | SS | MS | F | P |
| Store Number | 2 | 12.341 | 6.1707 | 27.60 | 0.000 |
| Item Number | 21 | 127.422 | 6.0677 | 27.14 | 0.000 |
| Error | 42 | 9.390 | 0.2236 | ||
| Total | 65 | 149.153 |
Model Summary
| S | R-sq | R-sq(adj) |
| 0.472821 | 93.70% | 90.26% |
Finding and Discussion:
In this experimental design we have used Store number and Item number as treatment and block. We get F-value of 27.60 and 27.14 respectively for Store number and block number. In this testing we define Null hypothesis as all treatments mean are equal. In contrast we define alternate hypothesis as: “at least two treatment means differ for treatment”.From the minitab, we get F-value of 27.60 and p value 0 for treatment. This rejects Null hypothesis and accept an alternate hypothesis. This conclude At least two treatment means (mean price for two store) are different
And furthermore, for block design we define Null and Alternate Hypothesis respetively. Null hypothesis is define as ” all block mean are equal”. Whereas alternate hypothesis is define as ” at least two block mean are different”.Similarly, for the block design also we get F-value as 27.14 and p-value as 0 which also conclude at least two block means are different.
Two sample t-test and Paired t-test:
Third and final test is two sample t test and paired t test. I have used data from two stores to compare the mean between the prices. They are Family Fare and Hornbachers. This test whether the mean price at two store are same or different. I will discuss the Minitab output in section below.
Result from Minitab:
When Equal Variance are Assumed
Two-Sample T-Test and CI: Family fare, Hornbachers
Method
| μ₁: mean of Family fare |
| µ₂: mean of Hornbachers |
| Difference: μ₁ – µ₂ |
Equal variances are assumed for this analysis.
Descriptive Statistics
| Sample | N | Mean | StDev | SE Mean |
| Family fare | 22 | 4.38 | 1.42 | 0.30 |
| Hornbachers | 22 | 4.45 | 1.73 | 0.37 |
Estimation for Difference
| Difference | Pooled StDev |
95% CI for Difference |
| -0.066 | 1.581 | (-1.028, 0.896) |
Test
| Null hypothesis | H₀: μ₁ – µ₂ = 0 | |||
| Alternative hypothesis | H₁: μ₁ – µ₂ ≠ 0 | |||
| T-Value | DF | P-Value | ||
| -0.14 | 42 | 0.890 | ||
When Equal Variance are not Assumed
Two-Sample T-Test and CI: Family fare, Hornbachers
Method
| μ₁: mean of Family fare |
| µ₂: mean of Hornbachers |
| Difference: μ₁ – µ₂ |
Equal variances are not assumed for this analysis.
Descriptive Statistics
| Sample | N | Mean | StDev | SE Mean |
| Family fare | 22 | 4.38 | 1.42 | 0.30 |
| Hornbachers | 22 | 4.45 | 1.73 | 0.37 |
Estimation for Difference
| Difference | 95% CI for Difference |
| -0.066 | (-1.030, 0.898) |
Test
| Null hypothesis | H₀: μ₁ – µ₂ = 0 | |||
| Alternative hypothesis | H₁: μ₁ – µ₂ ≠ 0 | |||
| T-Value | DF | P-Value | ||
| -0.14 | 40 |
|
||
- Paired t test
Paired T-Test and CI: Family fare, Hornbachers
Descriptive Statistics
| Sample | N | Mean | StDev | SE Mean |
| Family fare | 22 | 4.381 | 1.418 | 0.302 |
| Hornbachers | 22 | 4.447 | 1.730 | 0.369 |
Estimation for Paired Difference
| Mean | StDev | SE Mean | 95% CI for μ_difference |
| -0.066 | 0.476 | 0.101 | (-0.277, 0.145) |
µ_difference: mean of (Family fare - Hornbachers)
Test
| Null hypothesis | H₀: μ_difference = 0 |
| Alternative hypothesis | H₁: μ_difference ≠ 0 |
| T-Value | P-Value |
| -0.65 | 0.522 |
Finding and Discussion:
In this section we have compared the mean price of two stores using two sample t test (assume equal variance) and paired t-test. We have used mean prices for Family fare and Hornbachers. We have tested Null hypothesis as mean price is equal for two store vs mean prices are different as alternate hypothesis. Here we get p-value of 0.89 which is greater than alpha value (0.05) which doesn’t reject null hypothesis. Which means mean price for two stores are same and confidence interval also shows difference in mean can be equal to zero. Whereas using paired t -test, we tested mean difference equal to zero vs mean difference not equal to zero. Here we get p-value of 0.522 which means accept null hypothesis and mean difference is equal to zero. So result from both two sample t test and paired t test are same.
Data:
| Item Number | Item Name | Size | Family fare price | Hornbachers | Walmart |
| 1 | Kellog choclate frosted flakes | 13.7 oz | 4.69 | 4.49 | 4.39 |
| 2 | kellog frosted mini wheels blue | 24.3 oz | 4.19 | 4.19 | 3.88 |
| 3 | kellog raisin bran crunch | 15.9 oz | 4.49 | 4.09 | 2.53 |
| 4 | Kellog special K Red berrries | 16.9 oz | 4.99 | 5.99 | 3.88 |
| 5 | General mills Reesers puffs | 11.5 oz | 3.89 | 3.99 | 2.98 |
| 6 | kellog’s corn flakes | 18 oz | 4.99 | 4.49 | 3.17 |
| 7 | capin cruch (sweeted corn oat cereal) | 20 oz | 5.29 | 4.99 | 4.74 |
| 8 | quaker oats quick | 42 oz | 4.69 | 4.59 | 4.93 |
| 9 | natural valley cruncy granula bars | 8.94 oz | 3.39 | 3.79 | 2.89 |
| 10 | kellos fruit cracker hot wheels | 8 oz | 2.99 | 2.99 | 1.99 |
| 11 | Aunt jemina original pancake mix | 32 oz | 3.09 | 2.99 | 1.92 |
| 12 | Aunt jemina original syrup | 24 oz | 4.19 | 4.29 | 2.73 |
| 13 | Hershy syrup | 24 oz | 2.59 | 2.59 | 2.28 |
| 14 | coffe-mate nazelnut | 16 oz | 2.99 | 2.245 | 2.79 |
| 15 | Folgers house blend | 10.3 oz | 4.29 | 4.39 | 3.396 |
| 16 | betty crocker muffin chcoclate chip | 14.75 oz | 2.79 | 2.99 | 2.38 |
| 17 | welchs juice 100% grape | 64 oz | 4.99 | 4.89 | 3.84 |
| 18 | ocean spray cranberry 100% | 60 oz | 4.39 | 4.39 | 3.12 |
| 19 | ocean spary grape fruit juice ruby red | 64 oz | 3.49 | 3.49 | 2.98 |
| 20 | planters deluxe mixed nuts | 8.75 oz | 7.49 | 7.99 | 5.53 |
| 21 | blue diamond roasted salted almonds | 16 oz | 8.49 | 9.99 | 6.98 |
| 22 | cherrios cereal blueberry | 10,9 oz | 3.99 | 3.99 | 3.64 |
