Point Estimate of the Standard Deviation

Question

The National Football League (NFL) polls fans to develop a rating for each football game. Each game is rated on a scale from 0 (forgettable) to 100 (memorable). The fan ratings for a random sample of 12 games follow. 57 61 85 73 71 73 20 58 80 79 84 73

a. Develop a point estimate of mean fan rating for the population of NFL games (to 2 decimals).

b. Develop a point estimate of the standard deviation for the population of NFL games (to 4 decimals).

Answer 1

a. Develop a point estimate of mean fan rating for the population of NFL game (to 2 decimals)

Step 1: Define what a point estimate is:

A point estimate is a single value estimate for a population parameter. The most unbiased point estimate of the population mean 𝜇 is the sample mean 𝑥̅

Step 2: Use the formula: 𝑥̅=Σ𝑥/𝑛

Step 3: Find Σ𝑥

Σ𝑥=57+61+85+73+71+73+20+58+80+79+84+73=814

Step 4: Find n: n =12

Step 5: Calculate

𝑥̅ = Σ𝑥/𝑛=81412=67.83

The point estimate of mean fan rating for the population of NFL games (to 2 decimals) is 67.83

b. Develop a point estimate of the standard deviation for the population of NFL games (to 4 decimals).

Step 1: The best point estimate for the population standard deviation is s.

Step 2: Find variance 𝑠^2

Step 3: Find Σ𝑥2,Σ𝑥,𝑎𝑛𝑑 𝑛

Σ𝑥2=3249+3721+7225+5329+5041+5329+400+3364+6400+6241+7056+5329=58684

Σ𝑥=57+61+85+73+71+73+20+58+80+79+84+73=814

n=12

Step 4: Find 𝑠^2=315.2424

Step 5: Find s 𝑠=√𝑠2=√315.2424=17.7551

The standard deviation point estimate is 17.7551

(to 4 decimals)