Unit 6 : Inference for Proportions

H0

The old value that is hypothesized (assumed) to be true. Unless we get evidence otherwise, we assume this is still true.

H0: p = old/hypothesized value. This is also called the null hypothesis

In general, Ha will be one of :

Ha : P < hypothesized value

Ha : P > hypothezied value

Ha : P =/ hypothesized value

Test Statistic

Tells us how much evidence we have in favor of Ha (or how much evidence we have against Ho… how many Se’s the sample phat form the hypothesized value p

The test statistic is given by :

( sample statistic - null hypothesis value ) / SE

Sample statistic

the same estimate = phat

Hypothesized value

The H-0 value being tested = p

The average error between the sample result p^ and assumed true p is called

the standard error.

A large test statistic ~2 or ~3 means that

phat ia far from p, 2 or even 3 times the average amount of uncertainty due to randomness, so there is strong evidence in favor of Ha and we will likely reject H0

The p-value is the

tail area beyond the test statistic on the side specified by Ha.

The test statistic follows a __

standard normal distribution with mean = 0 and SD = 1. These are the default values for pnorm in R so we don’t need to include them

To get the p-value, use

pnorm (test statistic)

The P-value is the

tail area beyond the test statistic, on the side specified by Ha. This is the probability of getting a sample value the same as our current value or more extreme, under the assumption that H0 is true.

P VALUE < ALPHA :

Sufficient evidence to reject H0 (conclude Ha) —- IF THE P VALUE IS LOW, REJECT THAT HO!!!! (conclude Ha)

P-Value > ALPHA

Insufficient evidence to reject Ho (can’t conclude Ha), DO NOT REJECT THE HO

Small P-Value means

The observed sample result p^ is very unlikely. A small P-value means sufficient evidence to reject Ho and conclude Ha.

Large P-value means

The observed sample result p^ is quite reasonable, and the small difference could just be ordinary random variation. This large p-value means there is insufficient evident to reject H0 or to conclude Ha. Continue to give Ho the benefit of the doubt, but never ‘accept’ or ‘conclude’ Ho

If Ha points to the right, the P-value is the area on the right side. How do we compute this in R?

1-pnorm(test statistic)

Since normal distributions are symmetrical, the tail areas on the left and ride side are equal, so this means whenever Ha is one sided, we can use ___

pnorm(NEGATIVE TEST STATISTIC) to get the p-value

If Ha is =/, it is called

2-sided. This is DOUBLE the tail area on the smallest side.

To calculate p-value for a two sided Ha

2 x pnorm(negative value of test statistic)

The rejection threshold alpha is called the

significance level, and it is usually at 0.05, but sometimes 0.01

Alpha at 0.05 reduces what error

Type II

Alpha at 0.01 reduces what error

Typw I

In order for the test on a Normal Distribution to be valid, the sample size needs to be

at least np >= 10 and also n(1-p) >= 10.

How does increasing the sample size affect the P-value?

Larger sample sizes decrease the standard error, leading to larger test statistics, and smaller P-values, stronger evidence for Ha (easier to reject Ho)

Type I error

𝐻0 is really true, but a bad random sample leads
us to conclude 𝐻𝑎. This can be reduced by using 𝛼 = 0.01
(make it harder to reject 𝐻0).

Type II errror

𝐻𝑎 is really true, but a bad random sample leads
us to conclude 𝐻0. This can be reduced by using 𝛼 = 0.10
(make it easier to reject 𝐻0).

The critical value, z*

The value of the multiplier of standard errors that will produce the true P - within a normal curve. 95% of the time the true parameter p will be within 1.96 standard errors of the sample result phat

How to obtain z*

Use the function qnorm to find the cutoff value with 95% inside.

Since 0.95 is on the inside, there must be 0.05 outside, split equally with 0.025 above and 0.025 below. We could use qnorm 0.025 or qnorm 0.975. We discard the negative sign, it doesnt matter when doing +-

A confidence interval captures

the true p from the whole population

How should we interpret a confidence interval?

We are 95% confident the true proportion of ALL x who y is between (CI)

95% Confident means

we’re 95% confident in this procedure to successfully capture the true p. That is, the procedure’s success rate over many different samples is 95%

To increase the confidence level, you should increase

the critical value z*. A wider confidence interval has a greater chance of capturing the true p.

The margin of error, ME, gives

the maximum error between p and p hat, for the given confidence level. The wider the level, the greater the margin of error. More confudence implies a greater success rate in capturing the true value but this implies a greater margin of error and it might be too wide to be useful

CI & Hypothesis Tests

Since they’re two sided, they correspond with two sided hypothesis tests

Check to see if the Ho value is contained inside the interval :

If the Ho value is inside the interval we cannot reject it: there is insufficient evidence to reject Ho and we will have P-value > alpha

But if the Ho value is outside the interval, there is sufficient evidence to reject Ho, and P-value < alpha