Central limit for Sample means, t-tests

What is the difference between sample data of large and small n and actual population data represented on graphs?

small n for sample will be wider and have larger standard error Will have a distinct bell shape large n for sample will be narrower and have small standard error Will also have a distinct bell shape Real population will not look like a bell , it has some dips and curves

How do you calculate the standard error from the standard deviation given in minitab output?

st dev / sqroot(n)

x + /- [margin of error= z-statistic * (st. dev / sqroot(n)) ]

use this for sample means

There are two approaches for answering questions about a population mean...

1. Confidence intervals - Used for estimating parameter values 2. Hypothesis testing- used for deciding whether a parameter is one thing or another

What conditions must apply before using a CI for a population mean??

1. Random , INDEPEDENT sample 2. Large enough(30 or more) OR normally distributed (if less than 30) 3. Pop is 10x larger than sample

What 3 things to check to use CLT for sample means?

1. Random and independent 2. Large sample, At least 30 3. Larger population 10x larger than sample

Standard error?

= Standard deviation/ sqroot(n) Use this on minitab output and for everything in place of strd dev

If we were to use the data to construct the 95% confidence interval, would the interval be wider or narrower than the 90% confidence interval?

A 95% confidence interval would be wider than a 90% confidence interval because a 95% confidence interval would have a larger t* multiplier than a 90% confidence interval.

What is the t-statistic?

Hypothesis tests and confidence intervals for estimating and testing the mean are based on a statistic called the t-statistic t = (sample mean - population mean)/ (standard dev/sqroot(n))

A random sample of 35 college students was asked how many movies they had seen in the previous month. The sample mean was 4.14 movies with a standard deviation of 10.02. a. Construct a 90% confidence interval for the mean number of movies college students see per month. b. If we were to use the data to construct the 95% confidence interval, would the interval be wider or narrower than the 90% confidence interval? c. What would be the effect of taking a larger sample on the width of the interval?

N Mean StDev SE Mean 90% CI 35 4.14 10.02 1.69 (1.28, 7.00) a. We are 90% confident that the mean number of movies seen by college students in a month is between 1.28 and 7.00. b. A 95% confidence interval would be wider than a 90% confidence interval because a 95% confidence interval would have a larger t* multiplier than a 90% confidence interval. c. If we take a larger sample, the standard error would be smaller. This means the margin of error would be smaller and so the interval would be narrower.

Data on the speed (in mph) for random sample of 30 cars travelling on a highway was collected. The mean speed was 63.3 mph with a standard deviation of 5.23 mph. Find the 95% confidence interval for the mean speed of all cars travelling on the highway. Verify that the necessary conditions hold. Interpret the interval. Is it plausible that the mean speed of cars on the highway is 67 mph? Why or why not?

N Mean StDev SE Mean 95% CI 30 63.300 5.230 0.955 (61.347, 65.253) We are 95% confident that the mean speed of cars on the highway is between 61.35 and 65.25 mph. Since 67 is not in our interval, it is not a plausible value for the mean speed of cars on the highway.

The shape of the t-statistic depends on what?

The degrees of freedom df small the tails are thicker df large the tails are thinner

What is the big differenc betweeen t-statistic and z-statistic?

The t- statistic has a variable denominator[stand dev / sqroot(n)] The z- statistic has a denominator that is always the same (Stand Error)

When sample size is small you cannot use the normal distribution, so what must you use?

The t-statistic for sample below 30 or so

Suppose we take a random sample of 10 boys from this population. Can we find the approximate probability that the average weight of this sample will be above 89 pounds? If so, find it. If not, explain why not.

We have a random sample, the population is at least 10 times larger than the sample, and the population is Normal. So, we can apply the Central Limit Theorem even though the sample size is less than 25. The distribution of sample means is

Can you use the CLT for a sample size of 10?

YES but only if 1. Sample is said to be NORMAL 2. Sample is NOT skewed any which way 3. If skewed , would need a sample size to be at least 30!

What is the penalty for using data with such small sample size?

You must have a larger p-value, meaning there is less chance of rejecting the null.