Point Estimate:
An estimated single-point value for a population parameter.
Example: \(\bar{x} = 50\)

Interval Estimate:
An estimation range that contains the population parameter.
Example: \(50 \pm 2\), meaning the value is between 48 and 52.

Different types of Interval Estimation:

• Confidence
  

• Prediction
  

• Tolerance

  types explained in-depth in later slides

This example shows the point estimate (top line) and the interval estimate (bottom line).

pt <- 50
low <- 48
upper <- 52

ggplot() +
  stat_function(fun = dnorm, args = list(mean = pt, sd = 1), 
                linewidth = 1, color = "black") +
  
  geom_vline(xintercept = pt, linetype = "dashed") +
  annotate("text", x = pt + 0.2, y = 0.42, label = "mu", 
           parse = TRUE, vjust = -2.5, size = 4.5) +
  
  geom_segment(aes(x = 45, xend = 55, y = 0.05, yend = 0.05), 
               linewidth = 0.5) +
  geom_point(aes(x = pt, y = 0.05), color = "black", size = 5) +
  annotate("text", x = 52.5, y = 0.065, 
           label = "Point Estimate", hjust = 0, size = 4) +
...

...


  geom_segment(aes(x = 45, xend = 55, y = 0.02, yend = 0.02), 
               linewidth = 0.5) +
  geom_point(aes(x = c(low, pt, upper), y = 0.02), 
             color = c("blue", "black", "blue"), size = 5) +
  annotate("text", x = 52.5, y = 0.035, 
           label = "Interval Estimate", hjust = 0) +
  
  annotate("segment", x = low, xend = upper, y = 0.005, yend = 0.005, 
           arrow = grid::arrow(ends = "both", 
                               length = grid::unit(0.2, "inches"))) +
  annotate("text", x = pt, y = 0.001, 
           label = "\u00B1 2", size = 4.5) +
  
  coord_cartesian(xlim = c(45, 55), ylim = c(0, 0.45))

Confidence Interval (CI):
A confidence interval is a value range, derived from a set of statistics, that likely contains the true value of a population parameter.
It uses a single-point estimate and a margin of error (for stat variability) to express a confidence level, usually as a percentage (e.g. 90%).

Prediction Interval (PI):
A prediction interval is a value range, derived from a given population, that is likely to contain a new value.
Accounting for both variability and uncertainty, PI estimates where the new point will fall.

Tolerance Interval (TI):
A tolerance interval is a range of values, derived from sample data, that is likely to contain a specified portion with a given level of confidence.
This is useful for population distribution and spread.

When the Population Standard Deviation (σ) is Unknown

For a population mean \({\mu}\) with sample mean \(\bar{x}\), standard deviation \(s\), and sample size \(n\):

\[ \text{CI} = \bar{x} \pm t_{\alpha/2,\,n-1} \frac{s}{\sqrt{n}} \]

where:
- \(\bar{x}\) = Sample mean (point estimate)
- \(t_{\alpha/2,\,n-1}\) = Critical value from the t-distribution
- \(s\) = Sample standard deviation
- \(n\) = Sample size

When the Population Standard Deviation (σ) is Known

Z-distribution for when the \(\sigma\) is known:

\[ \text{CI} = \bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}} \]

where:
- \(\bar{x}\) = Sample mean (point estimate)
- \(z_{\alpha/2}\) = Critical value from the standard normal distribution.
- \({\sigma}\) = Population standard deviation
- \(n\) = Sample size

• As sample size (n) increases, the confidence interval width decreases because we can be more precise in our estimation in a bigger sample size.
• As sample size (n) decreases, the confidence interval width increases because there is more uncertainty in a smaller sample size.
  
• For a fixed n, higher confidence levels produce wider confidence intervals to ensure that the value lies within the interval.

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