Section 10.1 Estimating with Confidence AP Statistics February 11 th, 2011

An introduction to statistical inference Statistical Inference provides methods for drawing conclusions about a population from sample data. In other words, from looking a sample, how much can we “infer” about the population. We may only make inferences about the population if our samples unbiased. This happens when we get our data from SRS or well-designed experiments.

Example A SRS of 500 California high school seniors finds their mean on the SAT Math is 461. The standard deviation of all California high school seniors on this test 111. What can you say about the mean of all California high school seniors on this exam?

Example (What we know) Data comes from SRS, therefore unbiased. There are approximately 350,000 California high school seniors. 350,000>10*500. We can estimate sigma- x-bar as sigma/root 500=4.5. The sample mean 461 one value in the distribution of sample means.

Example (What we know) The mean of the distribution of sample means is the same as the population mean. Because the n>25, the distribution of sample means is approximately normal. (Central Limit Theorem)

Our sample is just one value in a distribution with unknown mean…

Confidence Interval A level C confidence interval for a parameter has two parts.  An interval calculated from the data, usually in the form (estimate plus or minus margin of error)  A confidence level C, which gives the probability that the interval will capture the true parameter value in repeated samples.

Conditions for Confidence Intervals the data come from an SRS or well designed experiment from the population of interest the sample distribution is approximately normal

Confidence Interval Formulas

Using the z table… Confidence level Tail Areaz* 90% % % or use the t-table at the back of the book

Confidence interval behavior To make the margin of error smaller…  make z* smaller, which means you have lower confidence  make n bigger, which will cost more

Confidence interval behavior If you know a particular confidence level and ME, you can solve for your sample size.

Example Company management wants a report screen tensions which have standard deviation of 43 mV. They would like to know how big the sample has to be to be within 5 mV with 95% confidence? You need a sample size of at least 285.

Review Section 5.2 Experimental Design AP Statistics February 11 th 2013

Statistical Significance An observed effect so large that it would rarely occur by chance is called statistically significant.

Double-Blind In a double-blind experiment, neither the subjects nor the people who have contact with know which treatment a subject received.

Experiments without placebos Matched pair design  In a matched pair design, subjects are paired by matching common important attributes.  Often the results are a pre-test and post-test with the unit being “matched” to itself.

Block Design A block is a group of experimental units or subjects that are known before the experiment to be similar in some way that is expected to affect the response to the treatments. In a block design, the random assignment of units to treatments is carried out separately within each block.

Section 10.2 Tests of Significance AP Statistics February 12 th 2013

The Test of Significance The test of significance asks the question:  “Does the statistic result from a real difference from the supposition”  or  Does the statistic result from just chance variation?”

Example I claim that I make 80% of my free throws. To test my claim, you ask me to shoot 20 free throws. I make only 8 out of 20. You respond: “I don’t believe your claim. It is unlikely that an 80% shooter makes only 8 of 20.”

Significance Test Procedure Step 1: Define the population and parameter of interest. State null and alternative hypotheses in words and symbols.  Population: My free throw shots.  Parameter of interest: proportion of made shots.  Suppose I am an 80% shooter This is a hypothesis, and we think that it is false. So we’ll call it the null hypothesis, and use the symbol H 0. (Pronounced: H-nought) H 0: p=.8  You are trying to show that I’m worse than a 80% shooter. Your alternate hypothesis is: H a : p<80%.

Significance Test Procedure Step 2: Choose the appropriate inference procedure. Verify the conditions for using the selected procedure.  We are going to use the Binomial Distribution:  Each trial has either success or failure.  Set number of trials.  Trials are independent.  Probability of success is constant.

Significance Test Procedure Step 3: Calculate the P-value. The P-value is the probability that our sample statistics is that extreme assuming that H 0 is true.  Look at Ha to calculate “What is the probability of making 8 or fewer shots out of 20?”  X is the number of shots made.  P(X<8)= =binomcdf(20,.8,8)

Significance Test Procedure Step 4: Interpret the results in the context of the problem.  You reject H 0 because the probability of being an 80% shooter and making only 8 of 20 shots is extremely low. You conclude that H a is correct; the true proportion is less than 80%. There are only two possibilities at this step  “We reject H 0 because the probability is so low. We accept H a.”  “We fail to reject H 0 because the probability is not low enough.”

Significance Test Procedure 1. Identify the population of interest and the parameter you want to draw conclusions about. State null and alternate hypotheses. 2. Choose the appropriate procedure. Verify the conditions for using the selected procedure. 3. If the conditions are met, carry out the inference procedure. Calculate the test statistic. Find the P-value 4. Interpret your results in the context of the problem

Example Diet colas use artificial sweeteners to avoid sugar. These sweeteners gradually lose their sweetness over time. Manufacturers therefore test new colas for loss of sweetness before marketing them. Trained tasters sip the cola along with drinks of standard sweetness and score the cola on a “sweetness score” of 1 to 10. The cola is then stored for a month at high temperature to imitate the effect of four months’ storage. Each taster scores the cola again after storage. What kind of experiment is this?

Example Here’s the data: 2.0,.4,.7, 2.0, -.4, 2.2, -1.3, 1.2, 1.1, 2.3 Positive scores indicate a loss of sweetness. Are these data good evidence that the cola lost sweetness in storage?

Significance Test Procedure Step 1: Define the population and parameter of interest. State null and alternative hypotheses in words and symbols.  Population: Diet cola.  Parameter of interest: mean sweetness loss.  Suppose there is no sweetness loss (Nothing special going on). H 0: µ=0.  You are trying to find if there was sweetness loss. Your alternate hypothesis is: H a : µ>0.

Significance Test Procedure Step 2: Choose the appropriate inference procedure. Verify the conditions for using the selected procedure.  We are going to use sample mean distribution:  Do the samples come from an SRS? We don’t know.  Is the population at least ten times the sample size? Yes.  Is the population normally distributed or is the sample size at least 25. We don’t know if the population is normally distributed, and the sample is not big enough for CLT to come into play.

Significance Test Procedure Step 3: Calculate the test static and the P- value. The P-value is the probability that our sample statistics is that extreme assuming that H 0 is true.  µ=0, x-bar=1.02, σ=1  Look at H a to calculate “What is the probability of having a sample mean greater than 1.02?”  z=(1.02-0)/(1/root(10))=3.226,  P(Z>3.226) = =normalcdf(3.226,1E99)

Significance Test Procedure Step 4: Interpret the results in the context of the problem.  You reject H 0 because the probability of having a sample mean of 1.02 is very small. We therefore accept the alternate hypothesis; we think the colas lost sweetness.

Assignment Exercises odd, odd Against All Odds Video Episode 20.