Someone makes a claim about the unknown value of a population parameter. We check
whether or not this claim makes sense in light of the evidence gathered (sample data)
• A test of statistical significance or hypothesis test tests a specific hypothesis using
sample data to decide on the validity of the hypothesis
• Example: You are in charge of quality control and sample randy 4 packs of cherry
tomatoes, each labeled ½ lbs (227 grams)
o The average weight from the 4 boxes is 222 grams
• Is the somewhat smaller weight simply due to chance variation?
• Is it evidence that the packing machine that sorts cherry tomatoes into packs
The t Test
1. State the null hypothesis (H 0
2. Decide on a one-sided or two-sided alternative hypothesis (H ) a
3. Choose a significance level a (optional)
4. Collect sample data and calculate t test statistic
5. Find the area under the curve (using the t table or software)
6. State the P-value and conclusion
There are only two possible conclusions for any hypothesis test:
a. There is sufficient evidence to reject the null hypothesis in favor of the
b. There is insufficient evidence to reject the null hypothesis in favor of the
Null and alternative hypotheses
• The null hypothesis, H 0 is a very specific statement about a parameter of the
• The alternative hypothesis, H ,ais a more general statement that complements yet is
mutually exclusive with the null hypothesis.
• Example: Weight of cherry tomato packs:
o H : 0 = 227 g (µ is the average weight of the population of packs)
o H : a ≠ 227 g (µ is either larger or smaller than as stated in H 0
One-sides vs. two-sided alternatives
• A two-tailed or two-sided alternative is symmetric:
o H : a [a specific value or another parameter]
• A one-tailed or one-sided alternative is asymmetric and specific:
o H : a < [a specific value or another parameter]
1 Stat 1400
o H a µ > [a specific value or another parameter]
• What determines the choice of a one-sided versus two-sided test is the question we are
asking and what we know about the problem before performing the test. If the question
or problem is asymmetric, taen H should be one-sideda If not, H should be two-sided.
• We draw a random sample of size n from an N(µ, σ) population.
When s is estimated from s, the distribution of the test statistic t is a
t distribution with df = n – 1.
H :o= o
This resulting t test is robust to deviations from normality as long as the sample size is
(page 232 from the textbook for explanation)
P-value: The probabilit0, if H were true, of obtaining a sample statistic as extreme as the one
obtained or more extreme in the directaon of H .
• Tomato Example: The packaging process is normal.
2 Stat 1400
H 0 µ = 227 g vs. Ha: µ ≠ 227 g
o The average weight from your 4 random boxes is 222 g with standard deviation 5
o What is the probability of drawing a random sample such as yours, or even more
extreme, if H0is true?
(page 232 from the textbook for more information)
Interpreting a P-value
• Could random variation alone account for the difference between H and o0servations
from a random sample?
Small P-values are strong evidence AGAINST H and w0 reject H . The f0ndings are
Large P-values don’t give enough evidence against H and 0e fail to reject H . 0
Beware: We can never “prove H .” 0
Range of P-values
• P-values are probabilities, so they are always a number between 0 and 1.