Point Estimation and Population

Point Estimation of a Population

• Estimation is a tool in mathematics used to make inferences about a population using gathered data
• The two types of estimation are point estimation and interval estimation
• A sample is a part of a population and aims to describe the whole group
• Statistics are sample characteristics used to infer information about the population
• Point estimation is a type of estimation using a single value (a sample statistic) to infer information about the population
• A population includes all members of a specified group
• A parameter describes the characteristics used to describe a population
• Interval estimation is a range of numbers in which a population parameter lies, considering the margin of error

Formula for Sample Mean

• Sample Mean = (Summation of x) / n, where n is the number of items in the population

Confidence Interval for Population Mean

• Confidence level is the probability that the interval estimate will contain the parameter, assuming many samples are selected and the estimation process is repeated on the same parameter

Confidence Interval

• Confidence interval is a specific interval estimate of a parameter, determined using data from a sample and a specific confidence level
• The three common confidence intervals are 90%, 95%, and 99%

Z-score

• Z-score formula: 𝑍𝑎/2
• The term 𝑍𝑎/2 * (σ/√𝑛) defines the maximum error of estimate

Confidence Intervals and Z-scores

• 90% Confidence Interval: Z-score = 1.645
• 95% Confidence Interval: Z-score = 1.96
• 99% Confidence Interval: Z-score = 2.58
• The confidence level is the percentage equivalent of the decimal value of 1-α

Confidence Intervals: Example

• For a 99% confidence interval, α = 0.01 because 1 - 0.01 = 0.99 (or 99%)

Sample Standard Deviation

• The sample standard deviation, s, can substitute σ in confidence interval formulas when n≥ 30
• The standard normal distribution is used to find confidence intervals for means

T-Distribution

• The t-distribution must be used when the population standard deviation is unknown, and the sample size is less than 30; the sample standard deviation replaces the population standard deviation, and the variable is normally or approximately normally distributed

Properties of the T-Distribution

• T-distributions are similar to standard normal distributions
  • They are bell-shaped
  • Symmetrical about the mean
  • The mean, median, and mode are equal to 0, located at the center of the distribution
  • The curve never touches the x-axis
• T-distributions differ from normal distributions because the variance is greater than 1
• The t-distribution is a family of curves based on the degree of freedom, which relates to the sample size
• As sample size increases, the t-distribution approaches a normal distribution

T-Table

• Degrees of freedom (df) are the number of values free to vary after computing a sample statistic
• For a confidence interval for the mean, df= n-1

Formula For Confidence Interval of The Mean

• Formula for a specific confidence interval for the mean when σ is unknown
• x̄ - 𝑡𝑎/2 *(s/√𝑛) < μ < x̄ + 𝑡𝑎/2 *(s/√𝑛)
• Where df = n-1, s = sample standard deviation, and μ = population mean

Finding 𝑡𝑎/2 for Confidence Interval

• Find 𝑡𝑎/2 for a 99% confidence interval when the sample size is 20
  • Degrees of freedom = n-1 = 20-1 = 19
  • Find 19 in the left column and 99% in the row labeled confidence intervals
  • The intersection of the two gives the value for 𝑡𝑎/2 , which is 2.861 (refer to the t-table)

How to Summarize

• Use 𝑡𝑎/2 if σ is unknown and n
• Use Za/2 if σ is known and n≥30.
• Use Za/2 if o is unknown and n≥30.