Review for Quiz 6 - Tagged

Review for Quiz 6 - Section 6.1 and 6.2

Key Concepts

• Population: The complete set of individuals that we are interested in studying.

• Sample: A subset of the population that is actually examined and for which we collect data.

• Parameter: A numerical characteristic of the population that is typically unknown.

• Statistic: A numerical characteristic of a sample, which can vary between different samples.

• Statistical Inference: The process of using sample data to make estimates or draw conclusions about a population.

• Sampling Distribution: The distribution of a statistic based on different samples of the same size drawn from the same population.

Properties of Sampling Distributions

• The value of a statistic varies from one sample to another due to sampling distribution.

• To minimize bias, random sampling should be employed.

• To reduce variability, utilize a large sample size.

• Larger sample sizes yield smaller variability and a more normally shaped distribution.

• The mean of the sampling distribution of the sample mean (x) equals the population mean (µ).

• Sample means exhibit lower variability compared to individual observations.

• Sample means are more normally distributed than individual observations.

• The standard deviation of the sampling distribution of sample means is smaller than that of the population by a factor of √n, where n is the sample size.

• Central Limit Theorem: States that for a sufficiently large sample size, the sampling distribution of x approaches a Normal distribution.

• If the population distribution is Normal, the sampling distribution of x for all possible sample sizes n is also Normal.

Practice Questions

• Question 1: Determine the sample size n given that the sampling distribution of x has a standard deviation of 10 and the population standard deviation is 30.

• Question 2: Adjusting sample size affects the center of the sampling distribution: it will STAY THE SAME. Variability will DECREASE.

Questions and Concepts

Sampling Methods

• Question 3: Sam used two sampling methods: one with n = 25 and another with n = 100. The distribution with a smaller spread is associated with n = 100 (more samples lead to less variability).

Standard Error

• Question 4: The standard deviation of the sampling distribution (Standard Error, SE) is expressed as σ/√n.

  • n: Sample size

  • σ: Population standard deviation

Heights of Young Women

• Question 5: The heights are normally distributed with μ = 65.1” and σ = 2.6”.

  • a: 95% of heights range between approximately 60.5” - 69.7” (using 1.96 standard deviations from the mean).

  • b: For a sample of 6 young women:

    • i: Mean = 65.1”, SE = 2.6/√6 ≈ 1.06.” (Sketch the distribution)

    • ii: Probability that their mean height is ≤ 68” can be calculated using the Normal distribution.

    • iii: According to the 68-95-99.7 rule, 95% probability that the sample mean falls between approximately 62.9” and 67.3” (whole numbers: 63” and 67”).

Effects of Increasing Sample Size

• Question 6: As the sample size increases:

  • a: The shape of the sampling distribution of x becomes more normal.

  • b: The mean of the sampling distribution of x stays the same.

  • c: The standard deviation of the sampling distribution of x decreases.

Identifying Parameters and Statistics

• Question 7: In a sample of firms in Colorado:

  • a: The parameter is the mean number of employees (21).

  • b: The statistic is the sample mean (12).

Summary: The Sampling Distribution of x

1. The mean of the sampling distribution is equal to the population mean (μ) regardless of sample size.

2. The standard error or standard deviation is expressed as σ/√n, indicating that a larger sample size results in a SMALLER standard deviation.

3. The distribution becomes more normal as n increases, a property validated by the Central Limit Theorem. If a population is normally distributed, the sampling distribution remains normally distributed regardless of sample size.