Chapter 10 Section 2 .pptx

For a 99% confidence interval, we want the interval

corresponding to the middle 99% of the

normal curve

.

Z-score for 99%:

2.576

Critical Value for 99%:

0.5% OR .005

For a 95% confidence interval, we want the interval

corresponding to the middle 95% of the

normal curve

Z-score for 95%:

1.96

Critical Value for 95%:

2.5% 0R .025

For a 90% confidence interval, we want the interval

corresponding to the middle 90% of the

normal curve

.

Z-score for 90%:

1.645

Chapter 10 Section 2

The Reasoning of a Significance Test Example 10.7:

Diet colas use artificial sweeteners to avoid sugar. Colas with artificial sweeteners gradually lose 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 period of time, then each taster scores the stored cola. This is a matched pairs experiment. The reported data is the difference in tasters’ scores. The bigger the

difference, the bigger the loss in sweetness. 2.0 0.4 0.7 2.0 -0.4

2.2 -1.31.2 1.1 2.3

The sample mean __________ indicates a small loss of sweetness. 

Consider that a different sample of tasters would have resulted in different scores, and that some variation in scores is expected due to chance.

Chapter 10 Section 2

Does the data provide good evidence that the cola lost sweetness in storage?

To answer that question, we will perform a Significance Test. (More steps will be added!!)

1. Identify the Parameter.

Chapter 10 Section 2

2. State the NULL Hypothesis.

There is no effect or change in the population. This is the

statement we are trying to find evidence against. The cola does not lose sweetness.

_____________

State the ALTERNATIVE Hypothesis.

There is an effect or change in the population.

This is the statement we are trying to find evidence for. The cola does lose sweetness.

_____________ OR _______________ OR _______________

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Chapter 10 Section 2

3. Calculate a Statistic to estimate the Parameter.

Is the value of the statistic far from the value of the parameter? If so, REJECT the null hypothesis. If not, FAIL the null hypothesis.

4. Calculate the P-VALUE.

Chapter 10 Section 2

P-values and Statistical Significance

Suppose the individual tasters’ scores vary according to a normal

distribution with mean and = 1. We want to test the null hypothesis so we assume

So the sampling model for is approximately normal with mean 0 (zero) and standard deviation

Chapter 10 Section 2

Our sample mean, , was 1.02. Assuming that the null hypothesis is true, what is the probability of getting a result at least that large?

The probability to the right of is called the critical value.

The P-VALUE is 0.0006, meaning that we would only expect to get this result in 6 out of 10,000 samples. This is very unlikely, so we will REJECT the NULL hypothesis in favor of the ALTERNATIVE

Chapter 10 Section 2

If the P-value is small we say that our result is Statistically Significant. The

smaller the P-value, the stronger the evidence provided by the data.

How small is small enough? Compare the P-value to the value of the

Chapter 10 Section 2

If the P-Value is as small or smaller than Critical Value , we say that the data are Statistically Significant at level _____.

Hypotheses can be Rejected OR Failed To Be Rejected

or is a ONE -sided hypothesis because we are only looking at one direction, greater than or less than.

is a TWO-sided hypothesis because we are looking at two directions, greater than and less than.

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Chapter 10 Section 2

Tests for a Population Mean

To test a claim about an unknown population parameter:

P H A N T O M S

Step 2: State the null and alternative HYPOTHESIS (one-sided or two-sided) in words and symbols.

Null hypothesis ______ : a statement about a population, expressed numerically in terms of same parameter ( Like )

Alternative hypothesis ______: expresses the effect we hope to find evidence for

MUST decide whether H_a should be one-sided or two-sided. If you do not have a specific direction in mind in advance, use a Two- sided alternative.

Example 1 (one-sided):

Example 2 (two-sided):

*Please note: Steps 1 and 2 are often combined into one step.*

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

Steps 3 & 4:

Choose the appropriate inference procedure.

Verify the conditions for using the selected procedure.

Since is known, we will use a one-sample

z

test. Now we

check the conditions.

Does the data come from an SRS? How were the data

produced? (Random is very important!)

Is the sampling distribution of approximately normal? Is

the

sample size large?



Step 5:

If the conditions are met, carry out the inference

procedure.

If not, proceed with caution. (Depending on the situation.)

Calculate the test statistic (z-score).

s

Step 6

: Find the

P

-value. (Don’t forget about the sketch.)

Probability of obtaining a sample statistic that is at least as far

from the mean as the observed sample statistic,

if the null

hypothesis is true !!

The p-value represents the strength of the evidence

The less probable the observed outcome (statistic) is, the stronger the evidence that H_o is INCORRECT,

The smaller the p-value, the stronger the evidence

AGAINST

H

.

If a significance level has been set, we reject H

when the

p-value is

LESS THAN

.

Step 7: Make a decision. (State alpha value.)

Reject or fail to reject the null hypothesis?

 ALWAYS State decision in terms of ALTERNATIVE Hypothesis, Meaning…

Reject  Support

 Fail To Reject Do Not Support

Step 8: Interpret your results in the context of the problem.

Failing to find evidence against null hypothesis only means that the data are consistent with null hypothesis, not that we have clear

evidence that the null hypothesis is true. Therefore, our decision will always be to “reject hull hypothesis” or “Fail to Reject null hypothesis – you should never accept the null hypothesis !!

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o

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How To Conclude A Significance Test



When do I

Reject

OR

Think… “When P-value is

LOW

, Reject H

” 



When do I

Fail To Reject

…

OR

Think… “When P-value is

HIGH

, Support H

”

P

-

value

>

CriticalValue

(

a

)

P

>

a

Hypothesis Testing With A Known Population Standard Deviation 

1. A school administrator has developed an individualized reading-comprehension program for eight grade students. To evaluate this new program, a random sample of 45 eight-grade students was

selected. These students participated in the new reading program for one semester and then took a standard reading-comprehension

examination. The mean test score for the population of students who had taken his test in the past was 76 with a standard deviation

2. The pain reliever currently used in a hospital is known to bring

relief to patients in a mean time of 3.9 minutes with a standard

deviation of 1.14 minutes. To compare a new pain reliever with the

current one, the new drug is administered to a random sample of

40 patients. The mean time to provide relief for the sample of

At the bakery where you work, loaves of bread are supposed to weigh 1 pound. From experience, the weights of loaves produced at the bakery follow a Normal distribution with standard deviation = 0.13 pounds. You believe that new personnel are producing loaves that are heavier than 1 pound. As supervisor of Quality Control, you want to test your claim at the 5% significance level. You weigh 20 loaves and obtain a mean

weight of 1.05 pounds.

1. Identify the population and parameter of interest. State your null and alternative hypotheses.

3. Calculate the test statistic and the P-value. Illustrate using the graph provided.

Here are the Degree of Reading Power (DRP) scores for an SRS of 44

third-grade students from a suburban school district: