Model Answer: Bar Diagram and Pareto Diagram

STEP 1 - Given Data

STEP 2 - Solution

1. Bar Diagram

In a bar diagram, each brand is shown as a separate rectangular bar whose height equals its number of sales. Bars are of equal width with equal gaps.

• X-axis: Brand (Dell, Acer, Lenovo, HP)
• Y-axis: Number of Sales

Heights: Dell = 50, Acer = 100, Lenovo = 40, HP = 60

No. of Sales
110|
100|          ████
 90|          ████
 80|          ████
 70|          ████
 60|          ████             ████
 50| ████     ████             ████
 40| ████     ████    ████     ████
 30| ████     ████    ████     ████
 20| ████     ████    ████     ████
 10| ████     ████    ████     ████
  0|__████_____████____████_____████__
     Dell     Acer   Lenovo    HP
                Brand

2. Pareto Diagram

A Pareto diagram arranges categories in descending order of frequency (as bars) with a cumulative percentage line overlaid.

Step 1: Arrange in descending order and compute cumulative percentage

$$\text{Cumulative %} = \frac{\text{Cumulative Sales}}{250} \times 100$$

Step 2: Draw bars (descending) with cumulative curve

Sales                          Cumulative %
110|                                    _____ 100%
100| ████                          ___/       84%
 90| ████                      ___/           
 80| ████            ______●--/               64%
 70| ████        ___/    
 60| ████    ●--/         ████
 50| ████  /              ████     ████
 40| ●---/           ████ ████     ████ ████
 30| ████            ████ ████     ████ ████
 20| ████            ████ ████     ████ ████
 10| ████            ████ ████     ████ ████
  0|__████____________████_________████_████__
    Acer(40%)      HP(64%)     Dell(84%) Lenovo(100%)

The line rises steeply for Acer and HP (together $64%$ of all sales) then flattens, illustrating the Pareto principle that a few categories account for the majority of the total.

Result: Bars: Dell 50, Acer 100, Lenovo 40, HP 60. Pareto order Acer, HP, Dell, Lenovo with cumulative percentages $40%, 64%, 84%, 100%$.

Model Answer: Interval Estimation and 99% Confidence Interval

Definition of Interval Estimation

Interval estimation is a method of statistical inference in which we estimate a population parameter by constructing a range (interval) of values, together with a stated confidence level, within which the true parameter value is expected to lie. It is expressed as:

$$\text{Point Estimate} \pm \text{Margin of Error}$$

The endpoints of the interval are called confidence limits, and the associated probability (e.g., 99%) is the confidence coefficient.

STEP 1 - Given Data

• Sample size: $n = 10$
• Marks: $45, 55, 20, 54, 40, 55, 51, 35, 43, 46$
• Confidence level: $99%$
• Population is normally distributed; population variance unknown → use t-distribution

STEP 2 - Solve

Step 1: Sample Mean

$$\sum x_i = 45+55+20+54+40+55+51+35+43+46 = 444$$

$$\bar{x} = \frac{444}{10} = 44.4$$

Step 2: Sample Standard Deviation

$$\sum(x-\bar{x})^2 = 1068.4$$

$$s = \sqrt{\frac{1068.4}{n-1}} = \sqrt{\frac{1068.4}{9}} = \sqrt{118.711} = 10.895 \approx 10.90$$

Step 3: t-value

$$df = n-1 = 9, \quad \alpha = 0.01, \quad \alpha/2 = 0.005$$

$$t_{0.005,,9} = 3.250$$

Step 4: Margin of Error

$$E = t_{\alpha/2}\cdot\frac{s}{\sqrt{n}} = 3.250 \times \frac{10.895}{\sqrt{10}}$$

$$\frac{10.895}{3.1623} = 3.4453$$

$$E = 3.250 \times 3.4453 = 11.197 \approx 11.20$$

Step 5: Confidence Interval

$$CI = \bar{x} \pm E = 44.4 \pm 11.20$$

$$\text{Lower limit} = 44.4 - 11.20 = 33.20$$ $$\text{Upper limit} = 44.4 + 11.20 = 55.60$$

Final Answer

The 99% confidence interval for the population mean is $(33.20,; 55.60)$, i.e., $44.4 \pm 11.20$.

We are 99% confident that the true population mean marks lie between approximately 33.2 and 55.6.