CK-12 Probability and Statistics - Advanced

http://www.ck12.org Chapter 7. Sampling Distributions and Estimations

t=

x ̄−μ

s/

√

n

t=

3. 21 − 3. 16

0. 067 /

√

11

t≈ 2. 48

If the value oftfrom the sample fits right into the middle of the distribution oftconstructed by assuming the null
hypothesis is true, the null hypothesis is true. On the other hand, if the value oftfrom the sample is way out in the
tail of thet-distribution, then there is evidence to reject the null hypothesis. Now that the distribution oftis known
when the null hypothesis is true, the location of this value on the distribution. The most common method used to
determine this is to find aP-value (observed significance level). TheP-value is a probability that is computed with
the assumption that the null hypothesis is true.

TheP-value for a two-sided test is the area under thet-distribution withd f= 11 −1, or 10, that lies abovet= 2. 48
and belowt=− 2 .48. ThisP-value can be calculated by using technology.

Press2ND [DIST]Use↓to select 5.tcdf (lower bound, upper bound, degrees of freedom)

This will be the total area under both tails. To calculate the area under one tail divide by 2.

There is only a 0.016 chance of getting an absolute value oftas large as or even larger than the one from this
sample( 2. 48 ≤t≤− 2. 48 ). The smallP-value tells us that the sample is inconsistent with the null hypothesis. The
population mean differs from the estimated mean of 3.16.

When theP-value is close to zero, there is strong evidence against the null hypothesis. When theP-value is large,
the result form the sample is consistent with the estimated or hypothesized mean and there is no evidence against
the null hypothesis.

A visual picture of theP-value can be obtained by using the graphing calculator.

This formulat= x ̄−μ
s/

√

n

is similar to that used in computing thezstatistic with the unknown population standard

deviation(σ)being substituted with the sample standard deviation.