Stat 2120: Unit 6.4 Lecture Activity
Part 1: Sampling Distributions and Hypothesis Tests
Learning Objectives:

Understand the relationship between test statistics, p-values, and evidence against the
null hypothesis.
1.1)
If we notice that our sample estimate is different from the hypothesized value of the
population parameter, what are two potential explanations for the observed difference?
1) Random sampling: our sample estimate could be different from Ho due to random
sampling/chance
2) Ho is incorrect. The sample estimate is different enough that random sampling alone
cannot explain the difference
1.2)



From the following options, what are possible conclusions from a hypothesis test?
Reject the null hypothesis. Our data supports the alternative hypothesis.
Fail to reject the null hypothesis. Our data does not support the alternative hypothesis.
Accept the null hypothesis. Our data supports the null hypothesis.
1.3)
A smaller / larger test statistic (in magnitude) leads to more evidence against the null
hypothesis. Briefly explain.
1.4)
A smaller / larger p-value leads to more evidence against the null hypothesis. Briefly
explain.
Part 2: Critical Values
Learning Objectives:


Understand how to find the critical value for a hypothesis test and how to compare the
test statistic with the critical value to make a conclusion.
Understand the relationship between confidence intervals and hypothesis tests.
Another way to make a conclusion from a hypothesis test is to use critical values, instead of pvalues. If your test statistic is more extreme than the critical value, we reject the null
hypothesis.
2.1) Obtain the critical value, z*, of a hypothesis test where
H0: µ = 145 and Ha: µ ≠ 145,
with significance level, α = 0.02. Assume that σ is known and that the sample size is large
enough for the normal approximation to work.
If p-value ≤ alpha, reject H0
Find critical value of a two-sided test
Ask self if p-value is exactly equal to alpha, what is the corresponding value of the z-score
Z* = Norm.S.Inv (0.99)
= 2.3263
Amy z-stat ≥ 2.3262 or ≤ -2.3263 will have a p-value < 0.02
2.2) Obtain the z-multiplier, z*, for a 98% confidence interval. Compare the z-multiplier with the
critical value from question 2.1. What is an implication based on this comparison?
Z* = Norm.S.Inv (0.99)
= 2.3263
98% CL and 2-sided test and alpha = 0.02 have the same z*. (1-alpha)*100% confidence interval
and 2-sided test at alpha will give consistent conclusions.
2.3) Obtain the critical value, z*, of a hypothesis test where
H0: µ = 145 and Ha: µ > 145,
with significance level, α = 0.02. Assume that σ is known and that the sample size is large
enough for the normal approximation to work.
One-sided
CL = 98%
=NORM.S.INV(0.98)
Any z-stat ≥ 2.0357 will lead to provide ≤ 0.02 reject H0.
2.4) Obtain the critical value, z*, of a hypothesis test where
H0: µ = 145 and Ha: µ < 145,
with significance level, α = 0.02. Assume that σ is known and that the sample size is large
enough for the normal approximation to work.
-z* = NORM.S.INV(0.02) = -2.0537
Due to convention, critical values are reported as positive values.
Any z-stat -2.0357 will lead to provide p-value ≤ 0.02 reject H0.
2.5) Repeat question 2.1 with significance level, α = 0.1.
Two-sided
Z* = Norm.s.inv (0.95)
= 1.6449
Part 3: Statistical and Practical Significance
Learning Objectives:


Explain what statistical significance means in context.
Understand the difference between statistical significance and practical significance.
A study is done to investigate if the average daily hours spent working at a manufacturing plant
is less than 8 hours a day. The corresponding hypothesis test is carried out with α = 0.05. Based
on a large sample, the sample mean was found to be 7.998 hours a day.
3.1) The p-value for this hypothesis test is 0.016. Based on this information, one of your
classmates makes the following statement: “Since our p-value is less than the significance level,
we have evidence that the average daily hours spent working at the manufacturing plant is a lot
less than 8 hours per day.” Do you agree or disagree with the statement? Disagreement
When we reject the null we say we have a statistically significant result which means that
random sampling alone cannot explain the difference between our sample estimate and the
null hypothesis.
** Hypothesis testing seeks to rule out random sampling. It does not give the magnitude of the
difference.
3.2) Why would constructing a 95% confidence interval for the mean daily hours spent in the
manufacturing plant be useful here?
A confidence interval gives a range of positive value for a population mean