Chapter 8 confidence intervals Activity Roll a Real

Chapter 8 confidence intervals

Activity Roll a Real die 50 times creating a Dot Plot to keep track of your rolls Calculate the sample mean of your 50 rolls, do this on the calculator What is the true mean and St. Dev of a real die? What is an easy way to calculate this?

Is the mean 3. 5? § Construct a 95% confidence interval for the true mean of the die.

§ N: Name the Interval § A: Assumptions/Conditions § S: Stats from calculator § C: Confidence Interval § A: And § R: Result in Context

95% Confidence Interval. How many of these intervals captured μ, which we know to be 3. 5 Typically, about 1 student per class will not have μ captured in their interval. This follows the meaning of a 95% confidence interval.

What is the meaning of these intervals?

Meaning of a 95% C. I. § The meaning is NOT: § 95% of all rolls are between(___) & (___). § It is: § If this process were to be done repeatedly, about 95% of all intervals would capture the true mean of the die. In simplest terms: If you did what you just did(roll 50 times, get the mean, make an interval) like a million times, then 95% of those intervals you made would capture μ.

90% Confidence Interval. § Now on your calculator make a 90% confidence interval for the same data using Z*. See any differences? § Try a 99% interval? What happens?

How tall are boys high school basketball players? The heights in inches for 28 boys on Varsity team rosters from around the country were used for the sample. Typically the standard deviation for these heights has been known to be σ = 6 inches. The sample is shown below in the table. 72 70 68 69 71 70 76 73 71 69 67 70 70 71 77 74 71 72 69 76 70 68 69 72 73 76 71 70 Construct and interpret a 99% confidence interval to estimate the mean height.

99% Z interval for means ASSUMPTIONS: We have an independent random sample of 28 boys high school basketball player heights. The population of all players of this age is well over 10 x our sample size(280). Our sample size is NOT large enough for CLT but a box plot shows no outliers or extreme skew, so there is no reason to doubt the normal condition. Stats Don’t forget me σ=6 n=28 I am 99% confident that the true mean height of boys h. s. basketball player heights is between 68. 3” and 74. 2”, because I used a method that captures the true mean in 99 out of 100 attempts in repeated sampling.

Interpreting an Interval § How much do the weights of soda cans vary? A random sample of 40 “ 12 ounce” cans are collected and the soda inside is measured. A 95% confidence interval for the true population mean μ is 11. 2 to 13. 1 (a) Interpret the confidence interval in context. (b) Interpret the confidence level in context. (c) Based on this interval, what can you say about the contents of the bottles in the sample? What can you say about the contents of bottles in this population?

Interpreting an Interval § How much do the weights of soda cans vary? A random sample of 40 “ 12 ounce” cans are collected and the soda inside is measured. A 95% confidence interval for the true population mean μ is 11. 2 to 13. 1 (a) Interpret the confidence interval in context. I am 95% confident that the true mean weight of 12 ounce soda cans is between 11. 2 and 13. 1 ounces.

Interpreting an Interval § How much do the weights of soda cans vary? A random sample of 40 “ 12 ounce” cans are collected and the soda inside is measured. A 95% confidence interval for the true population mean μ is 11. 2 to 13. 1 (b) Interpret the confidence level in context. If we took many sample of 40 “ 12 ounce cans” and constructed confidence intervals of those samples then about 95% of the intervals will capture the true mean weight of “ 12 ounce cans”

Interpreting an Interval § How much do the weights of soda cans vary? A random sample of 40 “ 12 ounce” cans are collected and the soda inside is measured. A 95% confidence interval for the true population mean μ is 11. 2 to 13. 1 (c) Based on this interval, what can you say about the contents of the bottles in the sample? What can you say about the contents of bottles in this population? The sample mean of the 40 cans was 12. 15 Since the claim on the can of 12 ounces is contained in this confidence interval, there is no evidence to suggest that the true mean is not 12 ounces.

Write down on a piece of paper the age of this man. Write down your best guess.

I would say that I am 100% confident that this man’s age is between 0 and 110 years old. But this is a useless confidence interval. If I want to me more accurate with my interval I have to lower my confidence. A more realistic interval might have a conclusion as……. I am 90% confident that this man’s true age is between 26 and 38 years old.

Class of 2014 Class Estimates

Mean = 32. 5 Med = 31 S = 9. 997 A 95% CI gives (30. 2, 34. 8) A 99% CI gives (29. 4, 35. 6) Class of 2015 Class Estimates

This is Luke Wilson as Richie Tenenbaum in the movie, “The Royal Tenenbaums” from 2001. He was 30 years old when this movie was filmed.

What are some ways to shrink your interval? § Lower confidence. § Higher sample size.

Confidence intervals— Day 1 § Take your die—the one you made--and roll it 50 times. § Create a Dot Plot § Describe the distribution…CUSS Use your calculator to calculate the mean and St. Dev, make sure you use s for the St. Dev, NOT σ

95% T Interval Since we are using a die that you made, we must use a T interval. Why? Because, the true standard deviation σ is unknown. We had to calculate it. In real life and in most statistics that are not made up, a T interval is used.

§ N: Name the Interval § A: Assumptions/Conditions § S: Stats from calculator § C: Confidence Interval § A: And § R: Result in Context

95% T Interval for means Assumptions We have an independent random sample of 50 rolls of a fake die. Since our sample is more than 30 our normal condition is met by the CLT. (3. 13, 4. 29) I am 95% confident that the true mean of my fake die is between 3. 13 & 4. 29 because I used a method that captures the true mean in about 95 out of 100 attempts in repeated sampling.

Confidence intervals— Proportions § Roll your die 75 times to see the proportion of two’s that you get. § Write down the number of two’s that you get. Lets create a 90% confidence interval for the proportion of 2’s.

90% Z interval for proportions Assumptions We have an independent random sample of 75 rolls of my created die. Our sample size of 75 rolls meets the Normality requirement…. (______, _______) I am 90% confident that the true proportion of 2’s that appear with my created die is between _____ & _______, because I used a method that captures the true proportion in about 90 out of every 100 attempts in repeated sampling.

Proportion Z interval Mr. Pines played some more Super Mega Baseball over Thanksgiving break. Mr. Pines wants to estimate the proportion of Homeruns hit in this game that are 450 ft or more. Actually wrote them down Homeruns(in feet)

Proportion Z interval We need the sample(p-hat) of HR’s that were 450 or more. Lets circle them. Homeruns(in feet)

(a) Construct and interpret a 96% confidence interval to determine the true proportion of Homeruns that are 450 ft or more in this game. Assumptions 96% Z interval for proportions We have an independent random sample of 84 homerun distances hit in this game. Our sample size of 84 homeruns meets the Normality requirement…. (. 153, . 347) I am 96% confident that the true proportion of homeruns that are 450 ft or more in this game is between. 153 &. 347, because I used a method that captures the true proportion in about 96 out of every 100 attempts in repeated sampling.

Proportion Z interval (b) Hammer Longballo claims that more than 40% of HR’s in this game are 450 ft or more. What would you tell him? Since 40% is NOT contained in this interval, your claim may be incorrect.

Proportion Z interval (c) The margin of error for this confidence interval is much too wide. If we want the margin of error to be no more than 5% with 99% confidence, how many Homerun distances must we record? Use the most “conservative” estimate for p-hat should be. 50 n = 663. 6… We must record at least 664 homeruns.

How do we find the exact sample size we want? These are the margin of error formulas for Z and T.

Back to the REAL Die. How many times do we need to roll the die to have our CI accurate to within ±. 10 at 90% confidence? Use this formula After solving, n = 789. 26 So we would need at least 790 rolls.

Margin of Error vs Standard Error With any of the margin of error formulas the standard deviation part is called the standard error

Finding Z* Z* is called the critical value, the common critical values are 90, 95, and 99. You should memorize these. 90 = 1. 645 95 = 1. 960 99 = 2. 576 These can also be found on the table, see below

Finding Z* Z* can also be found on all graphing calculators using the inv. Norm function. Desired confidence level

Finding T* T* depends on the sample size and the degrees of freedom(df) df = n - 1 Example if we want 98% confidence with a sample size of 26, what do we use for T* Our df = 26 – 1 df = 25 Use the value where 98 and 25 meet

Finding T* The TI-83 calculators do not have the inv. T function, the TI -84 and TI-Nspire calculators do Similar to inv. Norm but you need to also include the df

Finding Z* and T* § In any case, you will always have the tables with you. Fastest way is look at the table.

What Critical t* Value would you use? § A 95% confidence interval based on n = 10 observations. § A 99% confidence interval from an SRS of 20 observations. § An 80% confidence interval from a sample of size 7.

How large a sample? § A laboratory scale is repeatedly weighing a 10 gram weight. The readings are Normally distributed and the Standard Deviation is known to be 0. 0002 grams. § How many measurements must be averaged to get a margin of error of ± 0. 0001 with 99% confidence?

How large a sample? § Mr. Pines will be driving around Orange County this weekend trying to estimate the true mean gasoline price advertised at gas stations for the Holiday season. Typically the standard deviation this time of year for gas prices is σ =. 03 § How many gas stations must Mr. Pines record prices for to have a margin of error of ± 0. 01 with 99% confidence?

Cutting the margin of error § A very common question is how much does your sample size have to increase in order to cut the margin of error in half? The sample needs to be 4 x as large.

TRY IT ON YOUR CALC A sample size 4 times as large cuts the margin of error in half

Reducing the margin of error § Making the margin of error ½ as large we had to multiply the sample size by 4. § It follows that…. . § If the desired the margin of error is to be 1/n as big, then the sample size needs to be multiplied by n 2

Reducing the margin of error § A poll taken at Rancho asked 45 students whether they are in favor of school uniforms. A confidence interval was constructed. § If they want to keep the same level of confidence but divide the margin of error in third, how many students will they need to have in the poll? 405

The Meaning of a CI § Explain what the meaning of a confidence interval is. You really need to work at this question. Lets say you did a 95% confidence interval. It means that if you took many samples and made a confidence interval for each sample, then 95% of those intervals would contain the true value.

The Meaning of a CI § Explain what the meaning of a confidence interval is. You really need to work at this question. Lets say you did a 95% confidence interval. It DOES NOT MEAN that there is a 95% probability that the true value is in this interval.

The Meaning of a CI § Explain what the meaning of a confidence interval is. You really need to work at this question. You need to beware when the word PROBABILITY is attached to a CI. The meaning of the % is how often your METHOD will capture the true mean.

Review Problem § In a recent survey of 1500 randomly selected U. S. adults, 68% of the respondents agreed with the statement “I should exercise more than I do. ” § (a) Construct and interpret a 96% confidence interval to estimate the proportion of the U. S. adult population that would agree with the statement. § (b) For this study, state one source of potential bias and how it would affect the estimate of the proportion of adults who would agree with the statement, “I should exercise more than I do. ”

Ch 8 Confidence Intervals MEANS PROPORTIONS Z interval or T interval Z interval always used

Which interval should we use? Z interval for means, T interval for means, or Z interval for proportions Sample was taken, they needed to know info about it, σ was unknown.

Which interval should we use? Z interval for means, T interval for means, or Z interval for proportions Sample was taken, however σ was known.

Which interval should we use? Z interval for means, T interval for means, or Z interval for proportions Proportion problem, only one option.

Which interval should we use? Z interval for means, T interval for means, or Z interval for proportions Sample was taken, they needed to know info about it, σ was unknown.

Which interval should we use? Z interval for means, T interval for means, or Z interval for proportions σ was known.

This is the formula for constructing a Z Interval by hand, we will not be doing this. We will use the graphing calculator only. However, it is important to be familiar with it for a MC type matching question.

This is the formula for constructing a T Interval by hand, we will not be doing this. We will use the graphing calculator only. However, it is important to be familiar with it for a MC type matching question.

This is the formula for constructing a Z Prop Interval by hand, we will not be doing this. We will use the graphing calculator only. However, it is important to be familiar with it for a MC type matching question.

Practice

Practice

Practice

99% Z interval for proportions Assumptions The 10, 904 students who responded to the survey can be considered an independent random sample. It is safe to assume that there at least 109, 040 undergraduate college students, 10% rule met. Our sample size of 10, 904 students meet the Normality requirement…. (______, _______) I am 90% confident that the true proportion of 2’s that appear with my created die is between _____ & _______, because I used a method that captures the true proportion in about 90 out of every 100 attempts in repeated sampling.