Sample and Population

• Population: is a collection of all items
• Sample: is a subset of population used as a representation of the entire population

Central Limit Theorem

• If (X₁, X₂, …, X_n) is a random sample of size n, drawn from any population with mean μ and variance σ², then the sample has:
  • Expected mean, μ
  • Expected variance, ^σ²/_n
• It forms a normal distribution:
  • X ~ N (μ, ^σ²/_n)

Examples

1. The weights of the trout at a trout farm are normally distributed with mean 1 kg and standard deviation 0.25 kg
   1. Find, to 4 decimal places, the probability that a trout chosen at random weighs more than 1.25 kg
      • Write down the distribution: X ~ N (1, 0.25²)
      • Write down the desired probability:
        • P(X > 1.25) = 1 – P(X < 1.25)
      • Standardize and evaluate:
        • 1 – P(Z < ^{1.25 – 1}/_0.25) = 0.1587
   2. If X̄ kg represents the mean weight of a sample of 10 trouts chosen at random, state the distribution of Ȳ, evaluate the mean and variance. Find the probability that the mean weight of a sample of 10 trouts will be less than 0.9 kg
      • Write down the distribution: X ~ N (1, 0.25²)
      • For a sample, the mean remains equal but the variance changes. Find the new variance:
        • Variance of sample = ^σ²/_n = ^0.25²/₁₀ = 0.00625
      • Write the distribution of the sample: Y ~ N (1, 0.00625)
      • Write the desired probability: P(Y < 0.9)
      • Standardize and evaluate:
        • P(Z < ^{0.9 – 1}/_0.00625)
        • = 1 – P(Z < ^0.1/_0.00625)
        • = 0.103

Point Estimate and Confidence Interval

• A point estimate is a numerical value calculated from a set of data (sample) which is used as an estimate of an unknown parameter in a population
• Examples of point estimates are:
  • Sample mean, x̄ estimates the population mean, μ
  • Sample proportion, ^r/_n estimates the population proportion, p
  • Sample variance, s² estimates the population variance, σ²
• Point estimates are close to the population value but are not the value
• We can determine a confidence interval where the population value is likely to lie in (x̄ – δ, x̄ + δ)

The Variance

• Variance can be calculated/given for either a sample or a population and there is a difference between them
• Using the divisor, n: This is appropriate when:
  • the data is given for the whole population and you are interested in the variance of the whole
  • the data is given for the sample and you are interested in the variance of just the sample
    
• Using the divisor (n – 1):
  • Appropriate to use when the data is given for a sample and you are estimating the variance of the whole population
  • The quantity of calculated, s², is known as the unbiased estimate of the population variance
    

Percentage Points for a Normal Distribution

• The percentage points are determined by finding the z-value of specific percentages
• For example, to find the z-value of a 95% confidence level, we can see that the 5% would be removed equally from both sides (2.5% on each side), so the z-value we would actually be finding would be of 100% – 2.5% = 97.5%
  
  

Confidence Interval for a Population Mean

• A Sample Taken from a Normal Population Distribution with Known Population Variance:
  
  • z is the value corresponding to the confidence level required and n is the sample size
  • The confidence interval calculated is exact
• Large Sample Taken from an Unknown Population Distribution with Known Population Variance:
  • By the Central Limit Theorem, the distribution of x̄ will be approximately normal, so the same method as above
    
  • The confidence interval calculated is an approximate
• Large Sample Taken from an Unknown Population Distribution with Unknown Population Variance:
  • As the population variance is unknown, you must first estimate the population variance, s, using the sample data
    
  • The confidence interval is an approximate

Examples

1. Heights of a certain species of animal are normally distributed with σ = 0.17 m. Obtain a 99% confidence interval for the population mean, with the total width less than 0.2 m. Find the smallest sample size required
   • For a 99% confidence interval, find z where Φ(z) = 0.995 (think of the 1% cut from both sides):
     • z = 2.576
   • Subtract the limits of the interval and equate to 0.2:
     
   • Substitute the information given and find n:
     • √n = ^0.2/_{2 × 2.576} × 0.17
     • n = 4126.53 ≈ 4130

Confidence Interval for a Population Proportion

• Calculating the confidence interval from a random sample of n observations from a population in which the proportion of successes is p and the proportion if failures is q
• The observed proportion of success p is ^r/_n, where r represents the number of successes
  

Examples

1. A random sample of n people were questioned about their internet use. 87 of them had a high-speed internet connection. A confidence interval for the population proportion having high-speed internet connection is 0.1129 < p < 0.1771.
   1. Write the mid-point of this confidence interval and hence find the value of n
      • Find the midpoint of the limits, finding p:
        • 0.1129 + ^{0.1771 – 0.1129}/₂ = 0.145
      • The mid-point is equal to the proportion of people with high-speed internet use, so:
        • ⁸⁷/_n = 0.145
        • ∴ n = 600
   2. This interval is an α% confidence interval. Find α
      • Using the upper limit, this was calculated by:
        • 0.1771 = 0.145 + z[√(87/600 × 513/600) ÷ 600]
        • ∴ z = 2.233
      • Use normal tables and find the corresponding probability
        • Φ(z) = 0.9872
      • The same area is chopped off from both sides of the graph, so:
        • 1 – 2(1 – 0.9872) = 0.9744
        • Hence α% confidence is 97.44%