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### SECTION 12.2

Tests About a Population Proportion

NOW WHAT

In this section we are interested in the unknown proportion, p of a population as opposed to the unknown mean of a population.

Keep in mind, p will have an approximately normal distribution, so it is

BACK TO THE WORLD OF z.

Our z statistic

- We don’t really know p for our standard deviation.
- So, when we do a test, replace p by p0.
- NOTE: When we did confidence intervals, we used in place of p instead of p0

Assumptions

- Data are an SRS from the population
- The population is at least ten times as large as the sample (which insures independence)
- A. For a significance test:
B. For a confidence interval:

The Steps for a One Proportion z-test

- State the hypothesis and name test
H0: p = p0

Ha: p ‹, ›, or ≠ p0

- State and verify your assumptions
- Calculate the P-value and other important values
- Done in calculator or…
- Using the formulas and tables

- State Conclusions (Both statistically and contextually)
- The smaller the P-value, the greater the evidence is to reject H0

Example

A coin is tossed 4040 times. There were 2048 heads. The sample proportion of heads is

= 2048/4040 = 0.5069

That’s a bit more than one-half. Is this evidence that the coin was not balanced?

Step 1—Parameter

- The population for coin tossing contains the results of tossing a coin forever. The parameter p is the proportion of all tosses that lands heads up. The null hypothesis says that the coin is balanced. The alternative hypothesis is two-sided, because we did not suspect before seeing the data that the coin favored either heads or tails.

H0: p = 0.5

Ha: p ≠ 0.5

Step 2—Conditions

- SRS—The tosses we make can be considered an SRS from the population of all tosses.
- Normality—Since np0=4040(.5)=2020
and n(1-p0)=4040(.5)=2020 are both at least 10, we are safe using Normal calculations

- Independence—Since we are sampling without replacement (?) we must have at least 40400 tosses in our population. That isn’t an issue.

Step 3—Calculations

- P-value ≈ 0.3783

Don’t forget to draw your curve. Remember, use p0 for your standard error calculations. Use this standard error when drawing the curve.

Step 4—Interpretation

- A proportion of heads as far from one-half (.5) as this one would happen about 38% of the time by chance alone, if the coin is balanced.
- For this reason, we would fail to reject the null hypothesis.
- There is virtually no evidence that the coin is unbalanced.
- As a reminder, this is not evidence that the null hypothesis is true. It is still possible the coin is unbalanced, we just don’t have strong enough evidence to convince anyone that it is unbalanced.

Using a Confidence Interval

- For the example of the coin, it is possible that the confidence interval would be more meaningful than the significance test. A 95% confidence interval is
(0.49152, 0.52235)

- We can see that 0.5 is plausible, but so are many higher proportions, including the proportion that we saw in our sample of 4040 tosses.

Another Example

- Publishing scientific papers online is fast, and the papers can be long. Publishing in a paper journal means that the paper will live forever in libraries. The British Medical Journal combines the two: it prints short and readable versions, with longer versions available online. It this OK with authors? The journal asked a random sample of 104 of its recent authors several questions. One question in the survey asked whether authors would accept a stronger move toward online publishing: “As an author, how acceptable would it be for us to publish only the abstract of papers in the paper journal and continue to put the full long version on our website?” Of the 104 authors in the sample, 65 said “Not at all acceptable.”
- Do the data provide good evidence that more than half of all authors feel that abstract-only publishing is not acceptable?

Step 1—Parameter

- The population of interest is all of the authors for this particular journal.
- The parameter is the proportion of these authors that disagree with the abstract-only printing of their articles
- The null hypothesis is that there will be an event split between those that oppose the abstract-only printing and those in favor.
- The alternative hypothesis is that more authors will be against the abstract-only printing.

H0: p = 0.5

Ha: p > 0.5

Step 2—Conditions

- SRS—The chosen authors were a random sample but not necessarily an SRS of all authors from this journal.
- Normality—Since np0=n(1-p0)=52 are both at least 10, we are safe using Normal calculations
- Independence—Since we are sampling without replacement we must have at least 1040 authors for this magazine in the population.

Step 3—Calculations

- P-value ≈ 0.0054

Don’t forget to draw your curve. Remember, use p0 for your standard error calculations. Use this standard error when drawing the curve.

Step 4—Interpretation

- Because of the small P-value, there is sufficient evidence to reject the null hypothesis.
- We can conclude that more than half of all authors from the British Medical Journal would be opposed to printing their articles in the abstract-only format.

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