Question: If the variance is greater than 1, the sale of the new product would be halted. The management further wants to compare the sales at the depot over the past 9 months shown in the table 2 below, on a quarterly basis. table 2 Month 1 2 3 4 5 6 7 8 9 Sales in millions 70 82 40 70 62 90 45 76 65 If the estimated sales in the 10th term are greater than 90 million shillings, the management will consider increasing the wages of their workers. Additionally the depot is officially supported to be opened four days a week. However, some residents have complained that its only opened twice a week. A quick survey reveal the following information in table 3 below. table 3 x 1 2 3 4 P(X = x) 0.15 0.35 0.41 0.08 Where: x represents the number of times the depot is open. P(X = x) represents the respective probabilities. If the expected number of times the depot is open is less than 3, the depot super will be changed. Some residents also complained about a particular soft drink that would make them drowsy. The probability that the drink could make someone drowsy was 0.25. A sample of 20 residents was investigated. If the probability that at least 8 of them get drowsy is greater than 0.5, the drink would be removed from the shelves. Task Based on numerical evidence, help the company management to make appropriate decisions. Item 6 The National volley ball championships are scheduled for this term. According to the organisers the weight of the learners is normally distributed with an average of 50 kg and a variation of 1. The sports department in your school has 50 learners to choose from randomly, and would like to take at least 12 students of only learners whose weight exceed 49 kg, then they will not participate. They need your help on the feedback to give the organisers. Additionally a group of 10 learners would like to move with the team as cheer leaders. The sports department has requested for the details of their ages and mass, which are shown in table 4 below. table 4 Age (yrs) 12 13 11 12 14 13 14 13 12 15 Weight (kg) 40 35 37 43 45 40 36 42 38 47 The department will consider taking the group, if the relationship between their age and weight is very strong. In the past, the learners had complained about the poor feeding by organizers during the National games, rating them as follows on a scale of 1-5. 5 rated the food 1, 13 rated it 2, 14 rated it 3, 8 rated it 4 and 20 rated it 5. The department would consider special meals for the learners during the championship if the variability exceeds 1.2. Task Using numerical evidence, help the sports department to make informed decisions.

1. **Problem Statement:** We need to analyze several statistical aspects based on the data provided to help management make decisions. 2. **Sales Variance and Quarterly Sales:** - Sales data for 9 months: $70, 82, 40, 70, 62, 90, 45, 76, 65$ (in millions). - Calculate variance to check if it is greater than 1. - Calculate quarterly sales (3 months each) and compare. - Estimate sales for the 10th month and check if it exceeds 90. 3. **Variance Calculation:** - Mean sales $\bar{x} = \frac{70+82+40+70+62+90+45+76+65}{9} = \frac{600}{9} = 66.67$ million. - Variance formula: $$\sigma^2 = \frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^2$$ - Calculate each squared deviation: $(70-66.67)^2=11.11$, $(82-66.67)^2=234.78$, $(40-66.67)^2=711.11$, $(70-66.67)^2=11.11$, $(62-66.67)^2=21.78$, $(90-66.67)^2=544.44$, $(45-66.67)^2=469.44$, $(76-66.67)^2=87.11$, $(65-66.67)^2=2.78$ - Sum: $11.11+234.78+711.11+11.11+21.78+544.44+469.44+87.11+2.78=2093.66$ - Variance: $$\sigma^2 = \frac{2093.66}{9} = 232.63$$ which is much greater than 1. - **Decision:** Since variance $>1$, the sale of the new product would be halted. 4. **Quarterly Sales:** - Quarter 1 (Months 1-3): $70+82+40=192$ - Quarter 2 (Months 4-6): $70+62+90=222$ - Quarter 3 (Months 7-9): $45+76+65=186$ - Quarterly sales comparison: Q2 > Q1 > Q3. 5. **Estimate 10th Month Sales:** - Using average monthly sales: $66.67$ million. - Since $66.67 < 90$, estimated sales for 10th month is not greater than 90. - **Decision:** No wage increase. 6. **Depot Opening Times Expected Value:** - Given $x = 1,2,3,4$ and $P(X=x) = 0.15, 0.35, 0.41, 0.08$ - Expected value: $$E(X) = \sum xP(X=x) = 1(0.15)+2(0.35)+3(0.41)+4(0.08) = 0.15+0.7+1.23+0.32=2.4$$ - Since $2.4 < 3$, depot supervisor will be changed. 7. **Soft Drink Drowsiness Probability:** - Probability a person gets drowsy: $p=0.25$. - Sample size: $n=20$. - We want $P(X \geq 8)$ where $X \sim Binomial(n=20, p=0.25)$. - Use normal approximation: - Mean: $\mu = np = 20 \times 0.25 = 5$ - Variance: $\sigma^2 = np(1-p) = 20 \times 0.25 \times 0.75 = 3.75$ - Standard deviation: $\sigma = \sqrt{3.75} = 1.936$ - Continuity correction: $P(X \geq 8) \approx P(Y \geq 7.5)$ where $Y$ is normal. - Z-score: $$Z = \frac{7.5 - 5}{1.936} = 1.29$$ - From Z-tables, $P(Z \geq 1.29) = 1 - 0.9015 = 0.0985$. - Since $0.0985 < 0.5$, the drink will not be removed. **Final decisions for company management:** - Halt new product sales due to high variance. - No wage increase as 10th month sales estimated below 90. - Change depot supervisor as expected opening times less than 3. - Do not remove soft drink as probability of drowsiness is low. --- **Slug:** sales-variance **Subject:** statistics **svg:** "" **desmos:** {"latex":"","features":{"intercepts":true,"extrema":true}} **q_count:** 1