Download 補充：假設檢定

假設檢定
問題

假設檢定時要用甚麼distribution來看?

Rejection region 一定在右邊嗎? 要如何
判別?

Rejection region & p-value的關係
假設檢定時要用甚麼
DISTRIBUTION來看?
Example 11.1
The manager of a department store is thinking about
establishing a new billing system for the store's credit
customers.
She determines that the new system will be cost-effective
only if the mean monthly account is more than $170. A
random sample of 400 monthly accounts is drawn, for
which the sample mean is $178.
The manager knows that the accounts are approximately
normally distributed with a standard deviation of $65. Can
the manager conclude from this that the new system will
be cost-effective?
11.
4
In Chapter 9, we know
We can generalize the mean and variance of
the sampling of two dice:
…to n-dice:
The standard deviation of the
sampling distribution is
called the standard error:
Based on CLT
The sampling distribution of the mean of a
random sample drawn from any population
is approximately normal for a sufficiently
large sample size.
The larger the sample size, the more closely
the sampling distribution of X will resemble
a normal distribution.
Hence,
If the population mean = 170, and we keep getting a
sample (size of 400) from this population, the sampling
distribution follows normal distribution approximately
and the mean = 170 and the standard deviation =
65/400^0.5
 HT(假設檢定) – 先假設(母體資訊)再檢定(樣本資訊)

◦ Assume population mean = 170 (H0 is true)
◦ Sampling dist. follows normal with mean 170, and std
65/400^0.5
◦ Get one sample from population
◦ Compare sample mean with sampling distribution
◦ Based on sample mean, we
 Reject H0 (if sample mean makes us to think the population mean is not 170)
 Do not reject H0 (if sample mean makes us believe population mean is 170)
Type I and Type II errors

Type I error
◦ If population mean is really 170, the sampling dist.
Mean = 170 and std. = 65/400^0.5
◦ Since it follows normal, it is still likely to get a sample
with an extreme (too large or too small) mean
◦ Type I error happens if we really get one sample from
the population with mean =170 but since the sample
mean is too large or too small, we mistakenly say that
the population mean is not 170 (reject H0)

Type II error
◦ If population mean is not 170, but the sample mean
we get is close enough to 170
◦ Hence, we say population mean is 170 (do not reject
H0)
REJECTION REGION 一
定在右邊嗎? 要如判別?
The system will be cost effective if the mean account
balance for all customers is greater than $170.
We express this belief as our research hypothesis, that is:
H1: µ > 170 (this is what we want to
determine)
Thus, our null hypothesis becomes:
這是我們關
心的
H0: µ = 170 (this specifies a single value for the
parameter of interest)
我們假設都是真的，
所以才可以用相關的
sampling distribution
COMPUTE
Example 11.1 Rejection region
It seems reasonable to reject the null
hypothesis in favor of the alternative if the
value of the sample mean is large relative to
170, that is if
>
.
我們關心是否sample mean > 170, 所
以若抓出來的sample mean太大，要
reject, 若關心的是是否sample mean
< 170, rejection region就會在左邊
α = P(Type I error)
= P( reject H0 given that H0 is true)
α = P(
>
)
11.
11
REJECTION REGION &
P-VALUE的關係

Rejection region是在設定的一個 下找一
個 threshold point，使得threshold point以
外的區域面積剛好是
◦ E.g.

= 0.05 , threshold point175.34
P-value是若你從這樣的一個sampling
distribution下能抓到的一個樣本平均為某
一數值及以上(如果是右尾)的機率是?
◦ 隱含抓到一個樣本且平均為175.34的p-value是
0.05
P-value=0.0069