STAT 9708 Chapter 10/11

Give an example statement about how to correctly interpret a CI.

"We are 95% confident that the interval captures the true population mean" "I am 95% confident that the interval from $100 to $140 contains the mean profit of all policies sold by this sales representative".

How do we calculate z-score for our sampling distribution of sample proportions?

(p-hat - p)/sqrt(pq/n)

How do you calculate a t-statistic?

(y-bar - mu)/SE(y-bar)

What are the conditions for valid confidence intervals (Means)?

1) Random sample 2) Normal condition (n >=30) - Or our parent population is also Normally distributed - Or parent population is roughly symmetric w/o skews, outliers. 3) Independence Condition (n <= 10% of population total)

What are the conditions for valid confidence intervals (For Proportions)

1) Random sample 2) Normal condition (n*p-hat >=10, n*(1-p-hat) >=10) 3) Independence Condition (n <= 10% of population total)

What are the critical values for these common confidence levels? 90%, 95%, 98%, 99%

90% = 1.645 95% = 1.96 98% = 2.326 99% = 2.576

What is a one-proportion z-interval? (Also known as one-sample z-interval)

A confidence interval for the true value of a proportion.

What is a confidence interval?

An interval of values found from data in such a way that a particular percentage of all random samples can be expected to yield intervals that capture the true population parameter.

As sample size increases what happens to the sampling distribution of sample statistic?

As sample size gets larger and approaches infinity we get closer to a perfect Normal distribution.

What is the margin of error?

Concept related to confidence intervals. It is the extent of the interval on either side of the estimate (the observed statistic value).

What is the Central Limit Theorem? (proportion version)

If a random sample is selected from a population with proportion p, and sample size n meets the normal condition, then p-hat ~ Approx N(mu(p-hat), sigma(p-hat)). The shape and distribution of population doesn't matter the above distribution will hold true.

What is the Central Limit Theorem? (mean version)

If a random sample is selected from any population with mean mu and variance sigma^2, and sample size sufficiently large (n>=30) then y-bar ~ Approx N(mu(y-bar), sigma(y-bar)

What is the sampling distribution model for a proportion?

If the Independence Assumption and Randomization condition are met and we expect at least 10 successes and 10 failures, then the sampling distribution of a proportion is well modeled by a Normal model.

What happens to our confidence interval if we increase sample size?

Increasing our sample size will make the confidence interval narrower, mainly because SE(p-hat) decreases as n-increases (n is in denominator).

Is it accurate to say there is 95% chance that this specific interval contains the true mean?

No because it implies that the mean may be within interval or somewhere else. This phrasing makes it seem as if the population mean is variable but its not. The interval changes from sample to sample but the population parameter we're trying to capture DOES NOT.

What is the Normal Condition for proportions?

Our sampling distribution of the sample proportions will be approximately normal in shape if: 1) np >= 10 AND 2) n(1-p) >= 10

In CI is our uncertainty about the interval or the true mean mu?

Our uncertainty is about the interval. The interval varies randomly, but the true mean is neither variable nor random, its just UNKNOWN.

How do you determine the sample size needed to calculate a CI for proportion with a specified MOE and confidence level?

Plug in values into our confidence level equation. p-hat +- [z* times (sqrt(p-hat(1-p-hat)/n)] In brackets is MOE 1) If we don't know the p-hat maximize numerator by using 0.5. 2) Calculate corresponding z* value for desired Confidence. 3) Solve for n sqrt(n) = z* times (sqrt(p-hat*(1-p-hat))/MOE)

What is Standard Error (SE) of the mean?

Same concept as for proportions but we use equation s/sqrt(n). s is the sample standard deviation

What is meant by degrees of freedom?

Sample size minus 1. Used for our students t distribution.

What is the std deviation of the sampling distribution of the sample mean?

Sigma (y-bar) = sigma/sqrt(n) sigma on the right side equates to the population std deviation.

Why do we use the students-t model for means?

Since we don't know the population standard deviation we need to use the samples standard deviation. Using the t-model will give us a slightly WIDER range and will actually capture the mu (mean) the correct # of times based on our Confidence Level.

What is meant by one sample t interval?

Taking one sample and constructing a CI based on that sample mean.

What is a Sampling Distribution?

The distribution of a statistic over many independent samples of the same size from the same population.

What is a confidence level?

The long-term success rate of the method of creating confidence intervals. How often this type of CI will capture the parameter of interest.

What is the mean of the sampling distribution of the sample proportion (p-hat)?

The mean is the true proportion value. Mean(p-hat) = p. This makes sense if you think about it -> The expected sample proportion is going to be the proportion of the population.

What does the t-model approach as n approaches infinity?

The normal model

What is meant by critical value?

The number of standard errors to move away from the estimate (mean of the sampling distribution) to correspond to the specified level of confidence.

What is Standard Error (SE) of proportion?

The standard deviation of the sampling distribution on the sample side uses the calculated statistics proportion.

What is the std deviation of the sampling distribution of the sample proportion?

The std deviation is the sqrt(p(1-p)/n) where p is the population proportion and n is the sample size. This only works if we know our population proportion, otherwise we use the sample proportion to calculate the standard error SE(p-hat).

How do you find the critical value?

z* is usually found by looking at a z-table or with technology.

How can you find a t-critical value?

Using a t-table or technology.

How do you determine the sample size needed to calculate a CI for mean with a specified MOE and confidence level?

We don't have df for t table and we don't know our sample standard deviation in this case. We fall back to z critical value and use z table and we are also HOPEFULLY given the population std deviation for reference. z * sigma/sqrt(n)

Does the interval change based on the actual sample proportion?

Yes - Our confidence interval changes depending on what our actual proportion is because we use our sample proportion to calculate our confidence interval.

How do you calculate the margin of error for a proportion?

z* times SE(p-hat) z-critical score times standard error of proportion

What percentage would you need to find in z-table or technology for a 94% confidence level?

Z score associated with 97th percentile. 3% --- 94% ---- 3%. Z* value will be 1.88

What is the general formula for a confidence interval of proportion?

estimate + or - Margin of Error

What are all the different meanings for the p's in statistics notations

p-hat - sample proportion P(X=10) - Probability p - population proportion

What is equation for confidence interval for means?

y-bar +- t-critical score [alpha/2, df] * SE(y-bar) (Section starting with t-critical score to end is the MOE)