AAEC Exam Three Study

Sampling Distribution of the Mean

Probability distribution of ALL possible values of the random variable computed from a sample size n from a population with mean mu and standard deviation sigma

Theorem

If a random variable X is normally distributed with mean mu and standard deviation sigma, then the distribution of the sample mean is normally distributed with mean alpha/x = alpha and standard deviation sigma_X[hat] = sigma/[square root]n

Shape

If the population is normally distributed, the distribution of X[hat] will be normal

Mean

The mean of the sampling distribution of X[hat] is alpha/x, which is equal to the mean of the population mu. Based on this result, X[hat] is termed an unbiased statistic (or estimate)

Statistic (X̄)

an unbiased estimate of the population parameter (μ) if the mean of ALL possible values of the statistic is equal to the corresponding parameter

Unbiased statistic

on the average, is equal to the corresponding parameter. Individually, a specific value of the statistic may (and generally is) different from the parameter

If infinity_X̄ = (X̄)/# = infinity then _____ is an unbiased statistic.

X[hat]

Standard deviation

is equal to the population standard deviation (σ) divided by the square root of the number of sample observations (n)

Standard error of the mean

the standard deviation of the sampling distribution of X[hat]

As the sample size increases, the sampling distribution of X[hat] becomes _____ and ______ and more concentrated around the mean. That is, as n increases, σ_X̄ = σ/[square root]n decreases

taller, thinner

What is the sampling distribution of the mean (X̄)?

the distribution of all possible sample means of size n from a population

What is the mean of the sampling distribution of X̄?

mu_X[hat] = mu (mean of sample means equals the population means)

What is the standard deviation of the sampling distribution of X̄ (standard error)?

sigma_X[hat] = sigma/[square root]n

What happens to the standard error as a sample size increases?

It decreases → sample means become less variable → better estimates of mu

Z-score for a raw value X

z = (X-mu)/sigma

Z-score for a sample mean X̄

z = (X[hat]-mu)/(sima/[square root]n)

Difference between P(X) and P(X̄)?

P(X) is individuals values (more spread out) while P(X̄) is sample means (less variability, tighter distribution)

Why are probabilities using X̄ difference from probabilities using X?

because X[hat] uses standard error (sigma/[square root]n), making the distribution narrower than the population

What z-value contains the middle 95% of a normal distribution?

+-1.96

What does “middle 95% of sample means” mean?

The range between mu_X[hat] +-1.96(sigma/[square root]n)

Shortcut formula for middle 95% of X̄

mu+-1.96(sigma/[square root]n)

What is the effect of increasing sample size (n)?

Decreases standard error → sample means cluster closer to mu → more accurate estimates

Difference between X̄ and X?

X[hat] uses sampling distribution (n matters, smaller variability) while X uses population distribution (no averaging, larger variability)

Central Limit Theorem

Suppose a random variable X has population mean mu and standard deviation sigma and that a random sample of size n is taken from this population. Then the sampling distribution of X[hat] becomes approximately normal as the sample size n increases. The mean of the distribution is alpha_X[hat] = alpha and the standard deviation is sigma_X[hat] = sigma/[square root]n

The population distribution can be _________ and have ______ shape.

non-normal, any

As the sample size _____, the distribution of X̄ approaches a ______ distribution.

increases, normal

For “____ n” (which assumed to mean n ≥ 30), X̄ will be approximately ______ distributed even if the population does not satisfy a “ “ distribution.

large, normal

The Central Limit Theorem focuses on the ______ of the distribution of the sample mean, i.e., is it normal?

shape

Statistical hypothesis

a statement or claim regarding a population parameter (e.g., mu=500)

Statistical test

a statistical procedure or decision rule that leads to establishing the truth or falsity of a statistical hypothesis

Null hypothesis H₀

a statement to be tested; assumed true until evidence indicates otherwise

Alternative hypothesis H₁

what is considered to be true if the null hypothesis is rejected; this is the claim that we seek evidence for

What is a hypothesis test?

a method for using sample data to decide whether to reject a claim about a population parameter

What type of test is μ<500 ?

Left-tailed test

What does a left-tailed test mean?

You are testing whether the population mean is significantly less than the hypothesize value

What is the test statistic formula when σ is known?

z = (X[hat]-mu)/(sigma/[square root]n)

What does the test statistic measure?

How many standard errors the sample mean is from the hypothesized mean

What is considered an “unusual” sample in hypothesis testing?

A sample that is very unlikely under H₀

What does it mean if P-value is large?

The sample result is not unusual under H₀ → do NOT reject H₀

What does it mean if P-value is very small?

The sample is very unlikely under H₀ → reject H₀

What is the decision rule in hypothesis testing?

Reject H₀ if the sample result is unusual (low probability under H₀)

What does z = -1 mean in this context?

The sample mean is 1 standard deviation below the hypothesized mean

What does z = -4 mean in this context?

The sample mean is 4 standard deviations below the hypothesized mean

Why is z = -4 strong evidence against H₀

because such an extreme value has near-zero probability if H₀ is true

What is the general idea of hypothesis testing?

Assume H₀ is true and check if the sample result is too extreme to be believable

What does it mean to “reject H₀”?

The sample provides strong evidence that H₀ is false

What does it mean to “fail to reject H₀”?

There is not enough evidence against H₀

Key idea about hypothesis testing logic

We do NOT prove H₁ true— we only test whether H₀ is unlikely

What role does probability play in hypothesis testing?

It measures how likely the sample result is if H₀ is true

We reject the null hypothesis if the sample mean is “too ____” standard deviations away from the null hypothesis (H₀) (i.e., an unusual event that happens <5% of the time)

many

A statistical hypothesis ALWAYS includes a _______ (e.g., H₀: μ=500), and never a _________ (x[hat]).

parameter, statistic

The null hypothesis (or simply hypothesis) ALWAYS includes an _______ sign.

equal

The __________ hypothesis includes <, >, ≠. Look for key phrases in the claim. For example, “more than” means >; “different from” means ≠; and “less than” means <.

alternative

The _________ hypothesis can be one-tailed or two-tailed

alternative

μ<500

One-tailed, left

μ>500

One-tailed, right

μ≠500

Two-tailed

If you are unsure whether to use a one-tailed or two-tailed hypothesis test, ALWAYS use a ____-tailed test.

two

α = significance level = probability of a Type ___ error (reject a true H₀)

I

β = probability of a Type ___ error (do not reject a true H₀)

II

When testing a statistical hypothesis, there is _______ a possibility that your conclusion will be wrong (Type I error or Type II error)

always

You can never say the null hypothesis is TRUE unless you have access to all the population data (and that never occurs). Rather, we say we ____ ____ ____ the null hypothesis.

do not reject

A ___________ result occurs when you reject the null hypothesis.

significant

P-value is the probability that the test statistics takes a value _______ equal to or more extreme than the value actually observed (in both directions for a two-tail test), assuming H₀ is true.

equal to or more

A “_____” P-value is evidence for H₀ and a “_____” P-value is evidence against H₀.

large, small

An extremely small P-value (<0.01) means that H₀ is ______ rejected or the result is ______ statistically significant.

strongly, highly

Three approaches for testing a statistical hypothesis

(1) classical

(2) P-value

(3) confidence-interval

Assumptions when using the classical method:

(1) the sample is obtained using simple random sampling (2) the population is normally distributed OR the sample size is “large” (n≥30)

What parameter is being tested in a Z-test for μ?

The population mean, mu

What is the null hypohesis in a two-tailed test?

H₀: mu = mu₀

What is the alternative hypothesis in a two-tailed test?

H₁: mu ≠ mu₀

What is the null hypothesis in a left-tailed test?

H₀: mu = mu₀

What is the alternative hypothesis in a left-tailed test?

H₁: mu < mu₀

What is the null hypothesis in a right-tailed test?

H₀: mu = mu₀

What is the alternative hypothesis in a right-tailed test?

H₁: mu > mu₀

What does α represent in hypothesis testing?

the level of significance

Common values for α?

0.10, 0.05, 0.01

What is a critical value?

The Z-value that separates the rejection and nonrejection regions

What is the rejection region (critical region)?

The set of all test statistic values where H₀ is rejected

Formula for the Z test statistic?

Z = (x[hat]-mu₀)/(sigma/[square root]n)

What does the Z test statistic measure?

The number of standard deviations the sample mean is from the hypothesized population mean

In a two-tailed test, when do you reject H₀?

When Z < -z(alpha/2) OR Z > z(alpha/2)

In a left-tailed test, when do you reject H₀?

When Z < -z(alpha)

What are the four main steps in hypothesis testing?

(1) state hypothesis

(2) choose alpha

(3) calculate test statistic

(4) draw conclusion

What does “fail to reject H₀” mean?

There is not enough evidence against the null hypothesis

What does '“reject H₀” mean?

There is sufficient evidence against the null hypothesis

“Less than”

Left-tailed

“Greater than”

Right-tailed

“Different from/not equal”

Two-tailed

A claim says the average battery life is different from 10 hours. What type of test is this?

Two-tailed test

A company wants to know if the average fill weight is less than 16 oz. What type of test?

Left-tailed

A teacher claims students score higher than 75 on average. What type of test?

Right-tailed test

Write the hypotheses for: “The mean is less than 50.”

H₀: mu = 50

H₁: mu < 50

Write the hypotheses for: “The mean is different from 8.”

H₀: μ = 8
H₁: μ ≠ 8

A calculated Z-score is -2.1 and the critical value is -1.645. Reject or fail to reject H₀?

Reject H₀

A calculated Z-score is 1.2 and the critical value is 1.645. Reject or fail to reject H₀?

Fail to reject H₀

In a two-tailed test, the calculated Z-score is -1.2 and critical values are +-1.96. Decision?

Reject H₀

In a two-tailed test, the calculated Z-score is -1.2 and critical values are +-1.96. Decision?

Fail to reject H₀

What happens to the rejection region when α gets smaller?

It becomes harder to reject H₀

What symbol represents the sample mean?

x[hat]