We looked at screen tension and learned that when we measured the screen tension of 20 screens that the mean of the sample was 306.3. We know the pop. standard deviation is 43. Find an 80% confidence interval for μ. Find a 99.9% confidence interval for μ. How large a sample would you need to produce a 95% confidence interval with a margin of error no more than 3?

Section 9.1 Introduction to Significance Tests

A Rose By Any Other Name Significance Tests go by a couple of other names: Tests of significance Hypothesis Tests

Inference So far, we’ve learned one inferential method: confidence intervals. Confidence intervals are appropriate when we’re trying to estimate the value of a parameter. Today, we’ll investigate hypothesis tests, a second type of statistical inference. Hypothesis tests measure how much evidence we have for or against a claim.

A significance test is a formal procedure for comparing observed data with a claim (also called a hypothesis) whose truth we want to assess. The claim is a statement about a parameter, like the population proportion p or the population mean µ. We express the results of a significance test in terms of a probability that measures how well the data and the claim agree.

The Reasoning Behind Tests of Significance I say I am an 80% free throw shooter. You say “PROVE IT!” So, I shoot 20 free throws and only make 8. You say that I’m a liar. Your reasoning is based on how often I would only make 8 or fewer free throws if I am indeed an 80% free throw shooter. What is this probability? In fact, this probability is 0.0001. The small probability of this happening convinces you that my claim was false.

Diet Colas Diet colas use artificial sweeteners, which lose their sweetness over time. Manufacturers test new colas for loss of sweetness before marketing them. Trained testers sip the drink and rate the sweetness on a scale from 1 to 10 (with 10 being the sweetest). The cola is then stored, and the testers test the colas for sweetness after the storage. The data are the differences (before storage – after storage), so bigger numbers represent a greater loss of sweetness. This is a matched pairs experiment!

Before 8.3 7.2 7 7.8 7.6 6.7 7.3 7.7 8 8.2 After 6.3 6.8 5 8.2 5.4 8 6.1 6.6 7.3 5.9

The Data 2.0 0.4 -0.4 2.2 -1.3 1.2 1.1 0.7 2.3 Most of the numbers are positive, so most testers found a loss of sweetness. But the losses are small, and two of the testers found a GAIN in sweetness. So… do these data give good evidence that the cola lost sweetness in storage? Start by finding x-bar.

Here’s our question: The sample mean is 1.02. That’s not a large loss. Ten different testers would likely give different sample results. Does the sample mean of 1.02 reflect a REAL loss of sweetness? OR Could we easily get the outcome of 1.02 just by chance?

Hypotheses We will structure our test around two hypotheses about the PARAMETER in question (in this case, the parameter is μ, the true mean loss of sweetness for this cola.) The two hypotheses are: the null hypothesis (no effect or no change) represented by H0 (H-naught) the alternative hypothesis (the effect we suspect is true) represented by Ha.

For the Cola problem In words, what is the null hypothesis? In words, what is the alternative hypothesis?

Our Hypotheses in symbols In this example,

What is a p-value? We’ve found p-values before. P-values are the area of the shaded region in our normal curve. Now that area has a name!!! A p-value is the probability of getting a sample result as extreme or more extreme given that the null hypothesis is true. The smaller the p-value is, the more evidence we have in favor of Ha, and against Ho. If the p-value is low (standard is less than .05), reject the Ho! When the p-value is low, we say the results are statistically significant.

Back to the cola Find the P-value for the problem. We find that the P-value is .0123. This means that only 123/10,000 trials would result in a mean sweetness loss of 1.02 IF the true mean is zero. Since this is so unlikely to happen, you have good evidence that the true mean is greater than zero.

Types of Alternate Hypotheses and Their Graphs These are called one-sided alternatives. You only shade “one side”. It is either greater than or less than. This is called a two-sided alternative. You shade “two sides”. It could be less than or greater than.

Another example Cobra Cheese Company buys milk from several suppliers. Cobra suspects that some producers are adding water to their milk to increase their profits. Excess water can be detected by measuring the freezing point of the milk. The freezing temperature of natural milk varies normally, with mean μ = -0.545°C and standard deviation σ = 0.008°C. Added water raises the freezing temperature toward 0°C, the freezing point of water. Cobra’s laboratory manager measures the freezing temperature of five consecutive lots of milk from one producer. The mean measurement is x-bar = -0.538°C. Is this good evidence that the producer is adding water to the milk?

Homework Chapter 9 #1-4, 14, 16, 17