Notes on Chapter 8: One Sample Inference

Chapter 8: One Sample Inference

Objectives

• To infer about a population mean through confidence intervals and hypothesis tests when the population standard deviation is unknown.

• To infer about a population proportion through confidence intervals and hypothesis tests.

Introduction to Inference Procedures

• The previous chapter introduced the basics of inference procedures for the population mean and population proportion.

  • Conditions for confidence intervals and hypothesis tests:

  1. The sample must be representative of the population (ideally from a simple random sample).

  2. The sample statistic must be normal:

     • By the Central Limit Theorem (CLT), normality occurs if sampling from a normal population or if the sample size is large (30 or more).

     • For proportions, normality occurs if the expected number of successes and failures is 10 or more.

  3. For population mean inference, the population standard deviation, σ, is known. (This third condition will be modified in Chapter 8, as it is often unrealistic.)

• No condition exists for working with the population proportion regarding known standard deviation.

Section 8.1: Realistic Inference for the Mean - The t-Distribution

• The confidence interval and hypothesis test for the mean introduced in Chapter 7 (the z-interval and z-test) require known population standard deviation σ.

• If σ is unknown, the sample standard deviation, s, can be used as an approximation, which modifies the formulas for confidence interval and test statistic:

  • The test statistic now becomes $t = rac{\bar{x} - ext{population mean}}{s / ext{sqrt}(n)}$.

• However, this new test statistic is not normally distributed.

• W. S. Gossett published a new distribution under the pseudonym ‘Student’, which is now known as the Student’s t-distribution.

Theorem for Student’s t-Distribution

• Assumptions:
  a) A random sample of size n is taken.
  b) The distribution of the random variable is normal or the sample size n is 30 or more.

• Thus, the distribution of $t$ is a Student's t-distribution with $n - 1$ degrees of freedom.

Explanation of Degrees of Freedom

• The formula for sample standard deviation is
  $s = rac{ ext{sum of squares}}{n - 1}$.

• The connection between the denominator and degrees of freedom comes from the calculation of standard deviation, specifically that the sum of deviations from the mean must equal zero.

  • For n data values, the first n-1 have degrees of freedom in choosing values, while the last value has no freedom to choose.

Graph of Student’s t-Distributions

• The Student’s t-distribution is a bell-shaped curve but is more spread out than the normal distribution.

• There are multiple t-distributions, one for each degree of freedom (df).

  • Graphically comparing t-distributions with df = 1 and df = 2 shows how they differ from the standard normal distribution.

• As the degrees of freedom increase, the t-distribution approaches the normal distribution. At n = 40, the difference between z and t distributions is minimal.

Finding Probabilities in t-Distributions

• Use technology (e.g., TI-83/84) to find probabilities for the t-distribution, utilizing the tcdf( ) command:

  • Example command: tcdf(lower limit, upper limit, df).

  • Example for t-values between -1.00 and 1.00 with df = 15: tcdf(-1.00, 1.00, 15), resulting in approximately 0.6668.

• The command invT(area to the left, df) can be used to find t-scores from given areas, similar to using the invNorm command in previous chapters.

  • Example: If area to the left is 0.90 and df = 15, run invT(0.90, 15) to find $t$-score.

Homework Assignments

1. Find the probabilities for the t-distribution for df = 15 over the specified limits.

2. Work through practical examples using the t-distribution standards.

Section 8.2: One-Sample Inference for the Population Mean

• Example scenario: Estimating mean height or salary from random samples.

• Confidence Interval Formula for One Population Mean (t-Interval):

  1. Define random variable and parameter.

  • $\bar{x}$ = sample mean, $ext{population mean}$.

  1. State and verify assumptions:
     a) The sample was representative.
     b) The sample mean is normally distributed by the CLT or large sample size.
     c) Since $ext{standard deviation}$ is unknown, use the t-distribution.

  2. Calculate the sample statistic and confidence interval, where:

  • $ext{CI} = \bar{x} ext{ ± margin of error}$.

  1. Interpret the confidence interval.

Example of Confidence Interval Calculation

• Example 8.2.1: A random sample of 20 IQ scores, find a 98% confidence interval.

  • IQ Scores provided for computation.

  • The T-distribution's critical value can be found using lookup tables or calculator commands.

Interpretation of Results

• The expression could be read as: “98% confident that the true population mean IQ lies between the calculated limits.”

Homework Assignments

• Practice confidence interval and statistical hypothesis testing scenarios through newly introduced examples, ensuring to check assumptions at every step.

Section 8.3: One-Sample Inference for the Population Proportion

• Confidence intervals for estimating proportions across varied scenarios (e.g. smoking rates among students).

• Outline for Confidence Interval for Population Proportion (1-Prop z-Interval):

  1. Define random variable and parameter.

  • $x$ = successes, $p$ = population proportion.

  1. Assumptions review: simple random sample with sufficient success counts.

  2. Compute sample proportions and confidence intervals (using the z-distribution critical values).

Homework Assignments

• Exercises include real-world studies estimating proportions under the conditions met for statistical validity, consistent with examples mentioned in sections.

Key Takeaways

• Utilize t-test and t-interval for inferences about single population means without known standard deviations.

• Perform hypothesis testing and confidence estimate for population proportions using established protocols in the chapter.