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Unit 5: Inferential Statistics

Population and Sample

In statistics, the Population is the entire set of items/individuals under study, while a Sample is a subset selected from the population.

  • Representative sample: An unbiased sample that captures the characteristics of the population.
  • Random Sampling: Drawing a sample from the population such that every unit has an equal chance of selection. Example techniques include Simple Random Sampling and Systematic Random Sampling.

Parameter and Statistics

  • Parameter: A numerical value summarizing some characteristic of the population (e.g., Population Mean \( \mu \)).
  • Statistic: A numerical value summarizing some characteristic of the sample (e.g., Sample Mean \( \overline{x} \)).

Statistical inferences use sample statistics to estimate or test hypotheses about population parameters. The Central Limit Theorem is pivotal here, stating that the sampling distribution of the sample mean approaches a normal distribution as the sample size gets larger, no matter what the shape of the population distribution.

Hypothesis Testing

A Hypothesis is an assumption made about a population parameter.

  • Null Hypothesis (\( H_0 \)): Typically the hypothesis of no difference or no effect.
  • Alternate Hypothesis (\( H_1 \)): The hypothesis that contradicts the null hypothesis.

The t-Test

Given a small sample size (usually less than 30), a t-test is used to determine if there is a significant difference between the means of two groups. The t-statistic is computed as:

\[ t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}} \]

where:

  • \( \overline{x} \) = Sample mean
  • \( \mu \) = Population mean
  • \( s \) = Sample standard deviation
  • \( n \) = Sample size
  • Degrees of Freedom (df) = \( n - 1 \)

Competency-Based Questions

Q1. A psychological study randomly samples 25 individuals out of a population of 10,000 to gauge reading speed. The population average reading speed is 220 words per minute. Is “220 words per minute” a parameter or a statistic? Explain your reasoning with respect to inferential statistics.

Q2. An LED bulb manufacturer claims their new bulbs have an average lifespan of 1000 hours. A consumer rights group tests 16 bulbs and finds an average lifespan of 970 hours with a standard deviation of 40 hours. State the Null and Alternate hypotheses.

Q3. With reference to Q2, compute the t-statistic for the sample data. Will you reject or not reject the null hypothesis at the given degrees of freedom? Frame your answer step-by-step.

Q4. Explain the difference between Simple Random Sampling and Systematic Random Sampling. Provide a scenario where systematic sampling might introduce a bias that simple random sampling avoids.

Answers

Ans 1. It is a parameter, because it describes the average of the whole population (the entire 10,000 individuals).

Ans 2. \( H_0: \mu = 1000 \) (The average lifespan is 1000 hours). \( H_1: \mu \neq 1000 \) (The average lifespan is not 1000 hours).

Ans 3. \( t = \frac{970 - 1000}{40 / \sqrt{16}} = \frac{-30}{10} = -3.0 \). With df = 15, the critical value at 5% significance level is approx 2.13. Since |-3.0| > 2.13, we reject the null hypothesis.

Ans 4. Simple random sampling gives every subset an equal chance. Systematic random sampling selects every \( k^{th} \) element. If the list has a periodic pattern (e.g., every 10th item is a manager), systematic sampling might only pick managers, introducing bias.