Statistics Final

Population

Entire group of interest

As the population standard deviation increases, the margin of error:

Increases

"Test Statistic"

Z

Suppose 1063 people out of a random sample of 2054 adults report that they play video games. What is the standard error of pˆ?

.0110

The Z distribution is a special normal distribution whose mean is ______ and standard deviation is ________.

0,1

About 68% of observations fall within _____ of the mean.

1 standard deviation

Criteria that need to be met in order for Binomial Distribution

1) Defined n 2) Same chance of getting the outcome each time 3) 2 possible outcomes 4) Set time 5) Trials independent of each other

Steps for solving for normal distribution

1)Draw a picture of the X distribution. Mark off the mean, and the 3 standard deviations on each side of the mean. 2)Write down the probability you are trying to find, using probability notation. Shade it in on the picture. 3)Change your probability for X to a probability for Z using the Z formula 4)Use the Z table to find the corresponding probability for Z 5)Use this to find the probability you want for X. (Your drawing will help!)

About 95% of observations fall within _______ of the mean.

2 standard deviations

Suppose you want to estimate the percentage of all OSU students who will take classes this summer. You take a random sample of 200 OSU students and find that 50 of them will take classes this summer. What is the statistic in this problem?

25%

About 99.7% of observations fall within ______ of the mean.

3 standard deviations

Suppose the time to serve a customer (X) has a normal distribution with mean 5 minutes and standard deviation 2 minutes. You want to investigate the 5% of customer service times that were the longest. What cutoff point are you looking for?

95th percentile for X

What is a sampling distribution?

A sampling distribution is the set of all values of a statistic for all possible samples of size n from the population.

Parameter

A single number summarizing the population

Parameter

According to the population

Statistic

According to the sample

The population mean:

Can be smaller than, or larger than, or equal to the sample mean.

"Estimate this parameter"

Confidence interval

The normal distribution is:

Continuous

f you take a random sample of 50 M&M's and record the number of M&Ms of each color, you have a binomial distribution. T/F

False

Let's say we have a random sample of n=61 and are testing a two sided hypothesis. The calculated z value is .04, is this sufficient evidence to reject the null hypothesis? Why or why not?

False (Z is the test-statistic, not the p-value. If Z is small, your data is close to the claim in Ho, so not much evidence against it.

If you are not able to prove the alternative hypothesis, this means the null hypothesis is correct. T/F

False (just means you didn't have enough evidence against it)29. The smaller α

If someone claims the population mean is 5 and you believe it's greater than 5, you write the following hypotheses:

False (x-bar is about data; need μ's)

If you want to estimate the population mean, which technique do you use?

Find a confidence interval for μ

If X is ABOVE the mean, the Z score will be...

Positive

Scatterplots examine relationships between what type(s) of variables?

Quantitative

P-value less than Ho

Reject the null, our Ha was correct

Statistic

Single number summarizing a sample

Standard Deviation=

Square root of the variance

Square root of np(1-p)

Standard deviation, Sigma

A z-value tells you how many ________ the x-value is above or below the _______.

Standard deviations, Mean

Correlations words

Strong, moderate or weak Positive or negative Linear

Sample

Subset of the population

The shape of the normal distribution is:

Symmetric

A p-value can change if you take a new sample.

TRUE (each sample gives a different test statistic and hence a different p-value)

To get the standard deviation,

Take the square root of the variance

If X has a normal distribution, when does X-bar also have a normal distribution?

The Central Limit Theorem is not needed; n can be any size.

Let X be the number of professional golfers that said YES to your survey question. What is the name of the distribution for X?

The binomial distribution

The Central Limit Theorem is important in statistics because it says:

The distribution of X-bar is approximately normal, no matter what the distribution of X is, as long as n is large enough.

Suppose you take a random sample of size n and look at the average X-bar. As n increases, which of the following statements is true?

The mean of X-bar stays the same.

As N goes to infinity...

The more likely it is to narrow to the actual population mean

pˆ=

The proportion of yeses in one sample

Which of the following is not a characteristic of a binomial distribution?

There is a set of n trials b. Each trial results in more than one possible outcome. The trials are independent of each other. Probability of success p is the same from one trial to another.

In lecture we learned that the mean of all the sample means equals the mean of the population. Explain WHY this equality makes sense.

This equality makes sense because our population is all possible values of X and our sampling distribution is the set of all possible samples which includes all possible values of X, n times.

A 95% confidence interval is wider than a 90% confidence interval if all else remains the same.

True

A large p-value means you have little evidence against Ho. T/F

True

P(Z < -2.74) = P(Z > 2.74)

True

There is a 100% chance that your sample mean lies within your 95% confidence interval T/F

True

We use statistics to estimate population parameters, not the other way around.

True

The smaller α (alpha) gets, the harder it is to reject Ho. T/F

True (alpha is your cutoff value to reject Ho)

We may get different conclusions for the same problem if hypotheses are different. For example: conclusion for H0: μ=8 and HA: μ<8 might be different with conclusion for H0: μ=8 and HA: μ≠8.

True (not-equal-to Ha has a doubled p-value)

Why do we have to check BOTH conditions rather than just one, to be able to use the normal approximation? (Including an example with your explanation is OK.)

We need to check both conditions because if our probability is large, np may be greater than 10 whereas n(1-p) may not be greater than 10. However, if our probability is small, n(1-p) may be greater than 10 whereas np may not be greater than 10.

If Z = 0 then where is X compared to its mean?

When Z = 0, then X = the mean.

X-bar is normally distributed if....

X is normally distributed OR X is not normally distributed but n>30 because CLT

If you roll a die 100 times and find the average, and I roll a die 200 times and find the average, who is more likely to have an average that is greater than 5?

YOU, Because you roll fewer times, your results have a higher chance to be further away from the mean (3.5). The more times you roll the closer to 3.5 your results are likely to be.

Which of the following is a test statistic?

Z=-2.67 (correct answer)

z score formula

Z=X‒μ ----- Where X=value you are trying to find σ out about, μ=the mean, and σ=standard deviation

What two ways can we do statistical inference?

a. Estimate a population parameter using information from a sample b. Test a population parameter using information from a sample

How many we expect in Binomial Distribution=

np

The mean of a binomial probability distribution with n trials and probability p of success is:

np

You can use the normal distribution to approximate a binomial distribution when the following conditions are met:

np ≥ 10 and n(1-p) ≥ 10.

There are two conditions that you have to check before using the normal approximation when X is binomial.

np≥10 and n(1‒p)≥10

If you get a new sample, which of the following elements of a hypothesis test will change?

p-value

Correlation of a sample (r) will always be a number between

-1 and 1

Suppose your confidence interval for the percentage of all American families planning a vacation for the summer is 30% to 40%. Now suppose the media reported that 50% of American families go on vacation during the summer, would you agree or disagree with them, based on your data?

Disagree

Goal of statistical inference

Estimate a test or population parameter using info from a sample

A p-value in hypothesis testing means the same thing as the sample proportion.

FALSE

Suppose your p-value in a hypothesis test is .055. Using the standards from this class what do you conclude?

Fail to reject Ho (for us, we use .05 as our cutoff value to reject Ho. This doesn't quite make it.)

Suppose your p-value in a hypothesis test is .055. Using the standards from this class what do you conclude?

Fail to reject Ho (since the p-value .055 > significance level of .05.)

How does the sample size effect the standard error?

Inversely, sample size up means standard error down

If X has some unknown distribution. What do we know about the distribution of X-bar?

It has an approximate normal distribution if the sample size n is large enough.

The Central Limit Theorem is important in statistics because:

It says for n ≥ 30, and any distribution that's not normal, the sampling distribution of Xis approximately normal.

As n increases, what happens to μx?

It stays the same

We collected some data and wanted to know if the data came from a normal distribution. We made a normal probability plot and it showed a straight line. What does this tell us?

It tells our data comes from a normal distribution.

What does the CLT apply to?

Just shape

P-value more than Ho

Keep the null, we were wrong

The Z distribution has a table that displays what type of probabilities:

Less than

Let X̄ be the sample mean from X's distribution. The standard error of X̄ is:

Less than the standard deviation of X

Correlation is a measure of the strength and direction of what type of relationship between two quantitative variables? _

Linear

Correlation is affected by outliers. Explain why, briefly.

Looking at the formula for r, correlation is based on the mean of X, the mean of Y, the SD of X, and the SD of Y. All four of these items are affected by outliers, as we learned in Chapter 1.

If X is below the mean, the Z score will be...

Negative

Correlation has ____________ units.

No

Does x̄ =X̄

No

Does x-bar=mu sub x?

No, mu sub x=mu sub x-bar

Suppose you want to estimate the percentage of all American families planning a vacation for the summer. Your confidence interval is 30% to 40%. What was your value of p-hat ?

Not enough info