Central Limit Theorem, combined

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Central Limit Theorem....

: For any population with a mean and standard deviation, the distribution of sample means for a sample size (N) will approach a normal distribution with a mean and standard deviation of standard dev/square root of sample size as N approaches infinity

(T/F) Obtaining a lot of samples and calculating their means is pretty close to representing the actual population

true

standard error equation

(σ / √n)

What can we do to increase power?

- increase sample size

2 Properties of Distribution of Sample Means

1. Mean sample =Mean population Mean 2. Normally distributed

Standard Error of the Mean tells....

: tells how well does a sample mean represent the mean

Which of these is a correct null hypothesis? equation A H 0: μ = 12 B H 0: μ > 12 c Ho: x(bar) = 12

A

Q4c State why, in answering part (b), you did not need to use the Central Limit Theorem.

A machine produces steel rods with LENGTHS THAT ARE NORMALLY DISTRIBUTED

Alternative Hypothesis

Ha

If Sample Size Increases and The Standard Error of the Mean Decreases, how does this represent the population mean?

Represents the population mean better

Sample Size Increases.....

The standard error of the mean decreases

Q8bii State why use of the Central Limit Theorem was required in calculating this confidence interval.

The taxi journey times are not known to be normally distributed

Q3c State why, in part (a), use of the Central Limit Theorem was not necessary.

The volume, in millilitres, of lemonade in mini-cans may be " ASSUMED TO BE NORMALLY DISTRIBUTED. "

Q11aiii State why use of the Central Limit Theorem is not required in answering part (a)(i).

The weight of sand in a bag can be MODELLED BY A NORMAL RANDOM VARIABLE

Q7aii State why, in calculating your confidence interval, use of the Central Limit Theorem was not necessary.

The weights of packets of sultanas may be ASSUMED TO BE NORMALLY DISTRIBUTED

Ho: A diet of fast foods has no effect on liver function What is two way Ha? One way Ha ?

Two: A diet of fast food affects liver function One: A diet of fast food increases liver function One: A diet of fast food decreases liver function

Q2c State, with justification, whether you made use of the Central Limit Theorem in constructing the confidence interval in part (a).

Yes. The volumes of the population is not known to be normally distributed. (No need to mention sample size here.)

Exponential Distribution

a continuous random variable (RV) that appears when we are interested in the intervals of time between some random events

Uniform Distribution

a continuous random variable (RV) that has equally likely outcomes over the domain, a < x < b; often referred as the Rectangular Distribution because the graph of the pdf has the form of a rectangle.

Normal Distribution

a continuous random variable (RV) with pdf where μ is the mean of the distribution and σ is the standard deviation.

Mean

a number that measures the central tendency; a common name for mean is "average." The term "mean" is a shortened form of "arithmetic mean."

X(bar) ~ N(μ, (σ / √n)) is....

sampling distribution of the mean

sample parameters,, mean: Sd:

x bar s

population parameters: mean: Sd:

μ σ

Standard Error of the Mean can be stated as:

"How far can we expect the mean to be from the mean?"

Ho: Growth rates of forest trees are unaffected by increases in carbon dioxide levels in the atmosphere. Two Ha: One Ha:

-Growth rates of forest trees are affected by increases in carbon dioxide levels in atmosphere - Growth rates of forest trees increase by increases in carbon dioxide levels in atmosphere - Growth rates of forest trees decrease by increases in carbon dioxide levels in atmosphere

2 Influencing Factors of Standard Error of the Mean

1. Amount of variability in the population 2. size of the sample.

Hypothesis Testing Steps

1. Check Assumptions 2. Hypotheses 3. use sample data to collect an estimate of that parameter 4. Compare your estimate to the claim (critical value) 5. Make a conclusion about the claim (reject or fail to reject)

Why does Mean 1 Not Equal Mean 2.

: because of random error and normal distribution

Normal distribution of Distribution of Sample Means

: regardless of shape of population mean, the distribution of means is ALWAYS normal.

Standard Error of the Mean equation is the same as the

: same as the Central Limit Theorem Equation

Which hypothesis should be written as an inequality? A the alternative hypothesis B the null hypothesis C either the alternative or the null hypothesis

A

Which of these is NOT a correct alternative hypothesis to correspond with H 0: μ = 8? A H a: μ ≠ 8 B H a: μ ≤ 8 C H a: μ > 8

B

Central Limit Theorem (CLT) tells us that for any population distribution, if we draw many samples of a large size, nn, then the distribution of sample means, called the sampling distribution, will:

Be normally distributed. Have a mean equal to the population mean, μ. Have a standard deviation equal to the standard error of the mean, σ / n‾ √σ/n

The Central Limit Theorem (CLT) tells us that for any population distribution, if we draw many samples of a large size, nn, then the distribution of sample means, called the sampling distribution, will:

Be normally distributed. Have a mean equal to the population mean, μ. Have a standard deviation equal to the standard error of the mean, σ/n‾√σ/n.

Which of these is NOT a correct null hypothesis? A H 0: μ 1 = μ 2 B H 0: μ 1 - μ 2 = 0 C H 0: μ 1 < μ 2

C

CLT Practical Rules Commonly Used #2

If the original population is itself normally distributed, then the sample means will be normally distributed for any sample size n (not just the value of n larger than 30)

Q9bi Explain the reason for the statistician's statement.

If you go three standard deviations below the mean, you're at - 115 minutes. This is impossible but three standard deviations below the mean is not unusual in a Normal Distribution.

Q6bi Explain why Y is unlikely to be normally distributed.

If you go two standard deviations below the mean, you're at - 13 minutes. This is impossible but two standard deviations below the mean is not unusual in a Normal Distribution.

Q12 Has the CLT been used anywhere in this question?

No. " The weight of rice in a packet may be modelled by a normal distribution. "

Q1 Are we using CLT in this question?

No. The length of one-metre galvanised-steel straps used in house building " MAY BE MODELLED BY A NORMAL DISTRIBUTION "

Q10bii State why the distribution of (Ybar), the mean of a random sample of 60 single visits, is approximately normal.

Since the sample size is over 30, the CLT applies.

As Variability of Population Increases....

Standard error of the mean increases.

CLT Conclusion #2

The mean of the sample means will be the population mean µ

CLT Given #1

The random variable x has a distribution (which may or may not be normal) with mean µ and standard deviation ø

Q9bii Give a reason why, despite the statistician's statement, your answer to part (a)(iii) is still valid.

The sample size is over 30 so the CLT applies

CLT Conclusion #3

The standard deviation of the sample means will approach (O/√n)

(T/F) Given a reasonably large sample size, the distribution of sample means from *any population is normal.

True NOTE: IT SAYS ANY NOTE: IT SAYS DISTRIBUTION OF *SAMPLE *MEANS

(T/F) Two sided Ha is preferred over one sided Ha. Explain why if true or false

True, because one sided Ha has to be justified, proven right!

Two way Ha is .... while one way Ha is...

Two way means that you are not taking a side, but one way means you are taking a side

Type II Error

When you fail to reject a false null hypothesis: you say your experiment didn't work but it did. (Beta denotes this error)

Type I Error

When you reject a true null hypothesis: your say your experiment worked, but it didn't (your alpha (a) is essentially the probability that type one error occurs)

If Variability of the Population Increases and Standard Error of the Mean Increases, how does this represent the population mean?

Won't represent the population mean as well

sampling distribution of the mean

X(bar) ~ N(μ, (σ / √n))

the average value of n independent instances of random variables from ANY probability distribution will have approximately a t-distribution when

after subtracting its mean and dividing by its standard deviation and the n is sufficiently large

If you hold everything else constant, then an increase in Type I error means a (decrease/increase) in Type II error

decrease

Ha (equal/ do not equal) while Ho (equal/ do not equal)

do not equal equal

(T/F) to reject null hypothesis, we want p > a

false; a > p

Central Limit Theorem, CLT

for any given population with a mean μ and a standard deviation σ with samples of size n, the distribution of sample means for samples of size n will have a mean of μ and a standard deviation of σ and will approach a normal distribution as n approaches infinity

Standard Error Z means

how far the sample mean from the population mean / how far are sample means in general from the population mean

As sample sizes increase, the graph of the skewed distribution gets...

normal; the larger the sample sizes the better

The standard error is usually smaller than the standard deviation of our (sample/population) but is usually equal to or close to our (sample/population)

population sample

What happens when the degrees of freedom increases

the peak starts to get closer to the peak of a standard normal distribution The tail starts to get thicker

If we calculate the 95% confidence interval, we would expect

the true population mean to be found in its range in 95% of our samples

Standard Error of the Mean wants...

to represent the population, and the confidence of that depends on it.

(T/F) Always want to assume your null hypothesis is true

true

(T/F) You NEVER accept the null hypothesis

true; you either reject the Ho or fail to reject the Ho

t distribution

when we have a sample and don't know the population variance

sample parameters: mean: Sd:

x bar s

Can statistical and research hypothesis equal?

yes

Are Hypotheses hard to prove that they are right?

yes; it's easier to prove something is wrong or NOT equal to what you think

3 Distributions

1. Population 2. Sample 3. Distribution of Sample Means.

variances

The differences between planned amounts and actual amounts

CLT Conclusion #1

The distribution of sample x̄ will, as the sample size increases, approach a normal distribution

Type I and Type II errors are (inversely/directly) related when everything else is held constant

inversely

dependent samples t-test

means of two conditions, when the same sample is used for both

Q6bii State why (Ybar), the mean of a random sample of 35 gas meter installations, is likely to be approximately normally distributed.

A sample size of over 30 means that the CLT applies.

The null and alternative hypotheses are written about A a population parameter. B sample data. C a sample statistic.

A; Ho and Ha are always about population parameters, however, when you're testing it out, you're looking at sample parameters

how would you describe a scenario where a < p?

Fail to reject null hypothesis: you are wrong There is not enough evidence to reject the null. There is not enough evidence to indicate that Ha is true

(T/F) you can accept the null but never do anything to Ha

False; You cannot accept the null but you never do anything to the Ha!!!

CLT Practical Rules Commonly Used #1

For samples of size n larger than 30, the distribution of the sample means can be approximated reasonably well by a normal distribution. The approximation get better as the sample size n becomes larger.

I DON'T UNDERSTAND THIS

Given H0 is true, the p-value of a test is the probability of getting a value as extreme or more extreme in the direction of HA. The size of our p-value will drive our conclusion about H0

Research hypothesis Reading Program X (RPX) will improve children's reading ability What is the Ha and Ho of a statistical hypothesis

Ho: Children using RPX will score the same on a comprehension test as children using the old reading program Ha: Ha: Children using RPX will have greater reading comprehension scores than children using the old reading program

You believe that men are more likely to play FPS video games than women. What is your Ho? What is your Ha?

Ho: Men and women play equal amounts of FPS video games (you want to collect data that shows this is wrong) Ha: Time playing video games for men > for women

Q10bi Explain why a normal distribution is unlikely to provide an adequate model for Y.

If you go two standard deviations below the mean, you're at - 31 minutes. This is impossible but two standard deviations below the mean is not unusual in a Normal Distribution.

Q14d The normal distribution provides a good model for many continuous distributions which arise in production processes or in nature. Explain why the Central Limit Theorem provides another reason for the importance of the normal distribution.

If you have a large enough sample size, then the means of those samples are approximately normally distributed regardless of the population's distribution.0

single sample t-test

a statistic to evaluate whether a sample mean statistically differs from a specific value

independent samples t-test

a test to determine if there is a difference between two separate, independent groups; conducted when researchers wish to compare mean values of two groups

Q5e Indicate where, if anywhere, in this question you needed to make use of the Central Limit Theorem.

a) No - route A journey times are normally distributed. b) No - route B journey times are normally distributed. c) Not relevant d) Yes - Car journey times are from an unknown distribution. Sample size of 36 journeys allows us to use CLT.

CLT

statistical theory that states that given a sufficiently large sample size from a population with a finite level of variance, the mean of all samples from the same population will be approximately equal to the mean of the population.

z test

test of the data compared to a general population with population parameters

sum of squares, SS

the sum of the squared deviation scores

(T/F) Z- scores only work for N(0,1)

true

population parameters,, mean: Sd:

μ σ

Hypothesis Testing Step: Check Assumptions

- Assumptions are requirements that must be met for the statistical test to be valid - Assumptions include: random sampling, independent observations

independent observations (Assumptions)

- has to be an independent observation, that is one observation doesn't affect others.

Null Hypothesis

- hypothesis that nothing happened - it didn't work - usually that something equals something else (or 0) - No difference between groups

Statistical Hypotheses

- set a claim and a counterclaim about a population parameter

a-value

- usually 5 % or 0.05 - if there is less than a 5% chance that the findings were due to chance, then the null can be rejected. - alpha is usually set by researcher, but conventionally 5% - WHAT PERCENT RISK ARE WE GOING TO TO TAKE

p-value

- what is the probability that we could get these values if the null were true. - want the probability to be very low

steps to developing a hypothesis:

1. Gather information (read and observe) 2. Research question 3. make research hypothesis 4. Make statistical hypothesis *Statistical hypothesis will more likely have numbers involved

Q13 Has the CLT been used anywhere in this question?

No. " The heights can be modelled by a normal distribution. "

Ho is

Null Hypothesis

Central Limit Theorem

Regardless of the distribution of a population, as n increases, the distribution of the *means of *random samples from the population will approach a normal distribution, specifically: N(μ, (σ / √n)) - sampling distribution of the means

CLT equation

Regardless of the distribution of a population, as n increases, the distribution of the means of random samples from the population will approach a normal distribution, specifically: N(μ, (σ / √n)) - sampling distribution of the means

how would you describe a scenario where a > p?

Reject null hypothesis: you are right Always use the word significant. so. There is a significant difference/finding/etc.

how are research hypothesis different from statistical hypothesis?

Research gives expected relationship between two variables, while statistical makes assumptions based off of a parameter. Compares Ha and Ho.

CLT Given #2

Samples all of the same size n are randomly selected from the population of x values.

standard error

is usually smaller than the standard deviation of our population,, but is usually equal to or close to our sample the standard deviation of a sampling distribution, simply put the standard deviation from a point

Power

probability of correctly rejecting the null hypothesis (1-B)

if small p-value ............. Ho if big p-value............. Ho

reject fail to reject

The goal is to (accept/reject) null hypothesis

reject rejecting proves that your experiment worked


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