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Estimation and Confidence Intervals

Estimation and Confidence Intervals

  • Understand the difference between point estimation and interval estimation.
  • Learn how confidence intervals provide a range of likely values.
  • Compute confidence intervals for means and proportions.
  • Visualize confidence intervals with probability distributions.

Point Estimation vs. Interval Estimation

Point estimation provides a single value as an estimate of an unknown population parameter.

  • Example: Using the sample mean \( \bar{x} \) to estimate the population mean \( \mu \).

Interval estimation gives a range of values where the parameter is likely to be found, with a confidence level.

Confidence Intervals

A confidence interval (CI) is an interval estimate of a population parameter. It has the form:

\[ \text{Estimate} \pm (\text{Margin of Error}) \]

The general formula for a confidence interval for the population mean \( \mu \) is:

\[ \bar{x} \pm z^* \frac{\sigma}{\sqrt{n}} \]

where:

  • \( \bar{x} \) = sample mean
  • \( \sigma \) = population standard deviation (or \( s \) if unknown)
  • \( n \) = sample size
  • \( z^* \) = critical value from the standard normal table

Common Confidence Levels and z-values:

Confidence Level z-value
90% 1.645
95% 1.960
99% 2.576

Derivation

From the Central Limit Theorem, the sampling distribution of \( \bar{x} \) follows a normal distribution:

\[ \bar{X} \sim N\left( \mu, \frac{\sigma}{\sqrt{n}} \right) \]

Rearrange the probability statement for a normal distribution:

\[ P\left( -z^* \leq \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} \leq z^* \right) = C \]

Multiplying through by \( \sigma/\sqrt{n} \):

\[ P\left( \bar{x} - z^* \frac{\sigma}{\sqrt{n}} \leq \mu \leq \bar{x} + z^* \frac{\sigma}{\sqrt{n}} \right) = C \]

This shows the confidence interval formula.

Visualization of Confidence Intervals

Examples

Example 1: A sample of 50 students has an average test score of 78, with a standard deviation of 10. Compute the 95% confidence interval for the population mean.

Using:

\[ \bar{x} = 78, \quad s = 10, \quad n = 50, \quad z^* = 1.96 \] \[ \text{Margin of Error} = 1.96 \times \frac{10}{\sqrt{50}} = 2.77 \] \[ CI = (78 - 2.77, 78 + 2.77) = (75.23, 80.77) \]

Interpretation: We are 95% confident that the true mean score is between 75.23 and 80.77.

Exercises

  • Question 1: A sample of 40 people has a mean height of 170 cm with a standard deviation of 8 cm. Find the 95% confidence interval for the population mean.
  • Question 2: A survey finds that 60% of 500 respondents support a policy. Compute a 99% confidence interval for the proportion.
  • Question 3: A sample of size 36 has a mean weight of 65 kg. Assume \( \sigma = 5 \). Compute a 90% confidence interval.
  • Question 4: A quality control study finds that 30 defective items are found in a sample of 200. Compute a 95% confidence interval for the defect rate.
  • Question 5: A researcher estimates the mean reaction time of a drug to be 2.5 seconds, with a sample of 100 trials and a standard deviation of 0.4. Find a 95% confidence interval.
  • Answer 1: \( CI = (167.5, 172.5) \).
  • Answer 2: \( CI = (0.553, 0.647) \).
  • Answer 3: \( CI = (63.63, 66.37) \).
  • Answer 4: \( CI = (0.107, 0.193) \).
  • Answer 5: \( CI = (2.42, 2.58) \).

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