Stat 217 – Day 15Statistical Inference (Topics 17 and 18)

Previously – Central Limit Theorem If taking random samples

from a population with proportion , and the sample size is large, then the sampling distribution of the sample proportions will follow a normal distribution with mean equal to the population proportion and standard deviation equal to

n

)1(

Activity 15-1 (p. 295)

Probability of sample proportion at least .25 = .156

In many, many random samples of American adults, about 16% of samples have > .25p̂

Activity 15-1

(g) What if sample size is 400 instead of 100?

A sample proportion .25 or larger is even more unlikely to come from a larger sample.

Activity 15-1

(i) What about the population size?

DOESN’T MATTER!

As long as it is much, much bigger than the size of the sample

(j) Virginia, = .209, probability of sample proportion exceeding .25?

SAME!

Preliminary Questions (p. 333) Which tire would you pick?

Left front Right front

Left rear Right rear

(driver)

Activity 17-1

1. Define parameter

Identify the symbol and specify the (unknown) parameter in words Make sure the type of number (mean, proportion),

the population of interest and variable are clear

2. Stating hypotheses

Ho: Ho-hum hypothesis Ho: parameter = hypothesized value

Ha: Aha! Hypothesis Ha: parameter <,>, ≠ hypothesized value

Good practice: always state in symbols and in words

Stating Hypotheses

Direction of alternative hypothesis depends on research question

Lab 1: Do infants prefer the helper?

(a) Ho: = .5

(b) Ha: > .5

3. Checking technical conditions Sample size condition Randomness condition

If sample size condition is not met? If randomness condition is not met?

Good practice: Include shaded and labeled sketch of the relevant sampling distribution

4. Test statistic

Standardize the observed statistic comparing it to the hypothesized value, dividing by the standard deviation of the statistic (amount of sampling variability)

5. p-value

Row-Row-Row your boat

It is key to knowWhat p-value means --

It’s the chance (with the null)you obtain data that’s

At least that extreme!

NOT the chance the null hypothesis is true!!

6. Stating conclusions

If you ask Ben a question he either says “no” or he says nothing.

So if he says nothing, does this “prove” he agrees?

Activity 17-1: Flat Tires (p. 334)(a) If nothing special, people will pick the right

front tire 25% of the time (proportion .25)

(b) parameter, (c) > .25

(d) 73(.25)=18.25>10; 73(.75)=54.75)>10

.051

.25

=34/73=.466p̂

24.4051.

25.466.

z

Less than a .02% chance would get a sample proportion this large if = .25 Strong evidence for > .25

2. H0: = .25

1. Let represent the population proportion that would pick right front

Ha: > .25

4. Test statistic

Table II: probability above < .0002 5. p-value

6. Reject H0

3. Technical conditions

In conclusion

6. We have strong evidence from this sample proportion, that more than 25% of the population would choose the right front tire in this situation Caution: This was not a random sample. We may

consider this sample representative of Stat 217 students in general at Cal Poly

Test of Significance (p. 338)

1) Define population parameter in words2) State 2 competing claims about parameter

null hypothesis H0: parameter = valuealternative hypothesis Ha: parameter < > or ≠ value

3) Check “technical conditions” for procedure4) Calculate test statistic (assuming H0 true)

comparing what observed to what conjectured in H0

5) Calculate p-value (see Ha for “more extreme”)how often see sample data this extreme when H0 true

6) Make a decision to either reject or fail to reject H0

Is p-value small?State conclusion in context

Compare p-value to “level of significance, ”

So what next?

If reject .25 as a plausible value for the , the proportion of all CP students who would pick the right front, next question might be what are plausible values for ?!

What about .3?

What about .4?

Observations

So we fail to reject .25 and .3 as a plausible value for Another reminder that when you fail to reject the

null, that doesn’t mean you have proven the parameter has that value.

but we reject .4 as a plausible value What is the set of all plausible values?

Which values of the parameter will we fail to reject based on our observed sample statistic?

The main idea

Plausible values of the population parameter are values that are not “too far” from the observed sample statistic

Say within 2 standard deviations When CLT applies, 95% of sample proportions

should fall within 2 standard deviations of the population proportion

Activity 16-1 (p. 312)

(a) Observational units Youths

(b) Variable Whether or not a television in their room

(c) Parameter or statistic Statistic, p-hat

(d) Want to know proportion of all American youth that have a TV in their room(e) Can’t determine it exactly (not a census)(f) But should be in the ball park (large random sample)

Activity 16-1

To estimate the standard deviation, use the sample proportion Standard error

(h) (.68(1-.68)/2032) = .0103 If we use p-hat (sample proportion) in place

of (population proportion), we obtain an approximate confidence interval for .68 – 2(.0103) to .68 + 2(.0103) (.659, .701)

Ta dah!

I am 95% confident that between 65.9% and 70.1% of all American youths have a television set in their bedroom

Why am I allowed to say this? Empirical rule Normal distribution Central Limit Theorem Random sample and n > 10 and n(1-)>10

Ok, use p-hat here too…

Test of Significance Calculator

To Turn in, with partner

(n) boys or girls CI (p. 318) Calculation and interpretation (I’m 95% confident

that…)

For Tuesday Activities 16-3, 16-6 (with technology)