Chapter 6

Question 1

Law of large numbers

Answer 1

As the sample size N increases, the sample mean (xbar) gets closer to the population mean (u)

-bigger= More accurate average
more flips = closer to true probability

Question 2

Gilvenko Cantelli theorem

Answer 2

As the sample size n increases, the sample distribution gets closer to the population distribution

Ex: imagine drawing 1000 stick figures-at first they look messy but as you keep drawing, they get closer to looking like a real person (the true population shape)

Question 3

Central limit theorem

Answer 3

As sample size N increases the sampling distribution gets closer to a normal distribution

we say N>30 is large enough to use unless the population is extremely skewed

Question 4

Linear combination of normal random variables

Answer 4

If the population is normally distributed, then the sampling distribution is normally distributed, no matter what n is

Question 7

Sample mean (x bar)

Answer 7

It is a statistic computed from a sample and used to estimate the population mean

Question 8

Sampling distribution

Answer 8

The probability of distribution of the sample across all possible samples of the population

Question 9

Meaning of the sampling distribution

Answer 9

-expected value of x bar is equal to the population mean

Question 10

Standard error

Answer 10

-the standard deviation of the sampling distribution

Question 11

Variability of x bar

Answer 11

Sample mean, varies from sample to sample, but we can quantify this variability and make inferences about the population mean

Question 12

Population distribution

Answer 12

-refers to the actual distribution of the actual values in the population. Pattern or shape of the entire data set that you’re drawing samples from. [Normal distribution, skewed distribution]

Question 13

Sample distribution

Answer 13

-distribution of the values within one specific sample drawn from the population, a single sample might not fully represent the population, especially if the sample is small

Question 14

Sampling distribution

Answer 14

-refers to the distribution of all possible sample means that could be obtained from samples of size N drawn from the population

Ex: if you took many different samples (1000 samples) each of size 10, and calculated the meaning for each sample, the sampling distribution would describe the distribution of those 1000 sample means

Ex; imagine you’re studying the weight of apples from a farm. The population distribution of Apple weights could be skewed, but there are a very few large apples that pull them in higher. If you take a small sample of apples, the sample distribution might not reflect the true pattern of Apple weights accurately. However, if you were to take many random samples of a larger size (like n = 100) the sampling distribution of the sample mean would tend to be more normal, even though the population distribution is skewed, the sampling distribution would still tend towards a normal shape as sample size increases.

Question 15

Summary

Answer 15

-if the population is normal, you do not need a large sample to approximate the population mean with the sample mean the sampling distribution of the sample mean will be normal regardless of the sample size

-If the population is not normal, larger samples are needed for the sampling distribution of the sample meaning to approximate a normal distribution. This is why larger samples are essential when the population distribution is skewed or non-normal.

-the sampling distribution of the sample mean gets closer to a normal distribution as the sample size increases, even if the population distribution is not normal.