Conducting Hypothesis Testing: Level of Significance and P-Value Calculation

In summary, the conversation discusses the use of significance levels and p-values in hypothesis testing. The first question compares the 10% and 5% significance levels and calculates a p-value to reject the null hypothesis. The second question considers the 1% and 5% significance levels and determines that the null hypothesis is rejected at the 5% level but accepted at the 1% level. The third question uses the same formula to calculate a value and suggests computing a p-value and choosing a significance level to determine whether to accept or reject the null hypothesis.
  • #1
twoski
181
2

Homework Statement



I have a few questions i have to solve. They are quite similar.

1. Given the following, draw conclusions about the 10% and 5% levels of significance. Then find a p-value.

[itex]n = 10,
mean: 8.179,
std. dev: 0.02,
|mean - μ| = 0.021,
z_{0.05} = 1.645,
z_{0.025} = 1.96,
H_{0}: μ = 8.2
[/itex]

Using the formula from my textbook, i compute a value to compare to [itex]z_{0.05}[/itex] and end up with 3.3203. Since this is greater than 1.645, we reject the null hypothesis. This value is also greater than 1.96 so we reject the null hypothesis in both cases.

Now, i have to compute a p-value. This is where it gets confusing because the textbook doesn't explain how i determine what sign ( ≤, ≥, >, < ) to use when i write it out. It seems to be completely arbitrary for all example questions i find. Is it based on the sign used in the null hypothesis?

This is my best guess:

P( mean ≤ 8.179 ) = [itex]\Phi(-3.32)[/itex] = 0.05%

2. Given the following, draw conclusions about the 1% and 5% levels of significance. Then find a p-value.

[itex]n = 64,
mean: 756.4,
std. dev: 33.9,
|mean - μ| = 8.9,
z_{0.05} = 1.645,
z_{0.01} = 2.33,
H_{0}: μ ≤ 747.5
[/itex]

I compute the value to compare to 1.645 and i get 2.1, which is greater so we reject the null hypothesis. At the 1% significance level, we see that 2.1 is less than 2.33 so we accept the null hypothesis.

For the p-value: P( mean ≥ 2.1 ) = [itex]\Phi(-2.1)[/itex] = 1.79%

Again, this is just going off of a similar example that used ≥ for whatever unexplained reason. Possibly because my null hypothesis uses a ≤?

3. Given the following, decide whether you accept or reject the null hypothesis.

[itex]n = 10,
mean = 26.4,
S^{2} = 12.2667,
std. dev ≈ S = 3.5,
|mean - μ| = 3.6,
H_{0}: μ ≥ 30
[/itex]

Using the same formula from the prior questions we end up with 3.25. We aren't given a level of significance so i guess I have to compute a p-value then come up with a level of significance on our own?
 
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  • #2
P( mean ≤ 3.6 ) = \Phi(-3.25) = 0.05%Since this is less than 5%, we reject the null hypothesis. Homework Equations Z = \frac{|mean - μ|}{std. dev.} P(\text{value} ≤ x) = \Phi(-Z)The Attempt at a Solution I think i have answered my own questions above but would like someone to look over it and confirm if i am on the right track or not.
 

1. What is the purpose of conducting hypothesis testing?

The purpose of conducting hypothesis testing is to determine whether there is enough evidence to support a particular hypothesis or claim. It helps scientists make informed decisions by analyzing data and determining the likelihood of the observed results occurring by chance.

2. What is the difference between a level of significance and a p-value?

A level of significance is a predetermined threshold that is used to determine whether the results of a study are statistically significant. It is usually set at 5% or 0.05. On the other hand, a p-value is the probability of obtaining the observed results or more extreme results if the null hypothesis is true. It is compared to the level of significance to determine if the results are statistically significant.

3. How is the level of significance determined?

The level of significance is typically set by the researcher based on the desired level of confidence in the results. A common level of significance is 0.05, meaning there is a 5% chance that the observed results occurred by chance. However, the level of significance can also be adjusted depending on the complexity and importance of the research.

4. How is a p-value calculated?

A p-value is calculated by determining the probability of obtaining the observed data or more extreme data if the null hypothesis is true. This is done by using statistical tests such as t-tests, ANOVA, or chi-square tests. The result is compared to the predetermined level of significance to determine if the results are statistically significant.

5. What does a p-value of less than the level of significance indicate?

If the p-value is less than the level of significance, it means that the observed results are statistically significant. This means that there is a low likelihood that the results occurred by chance, and there is enough evidence to reject the null hypothesis and support the alternative hypothesis.

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