Statistics and Probability: Sampling and Hypothesis Testing
What Is Sampling?
Sampling is the process of selecting a subset of individuals or items from a larger population to make inferences about the population as a whole. In statistics, it’s often not practical or possible to study an entire population, so samples are used instead.
There are different sampling methods that can be used, including:
- Random Sampling: Every individual in the population has an equal chance of being selected.
- Systematic Sampling: Individuals are selected at regular intervals from a list.
- Stratified Sampling: The population is divided into subgroups (strata), and samples are taken from each subgroup.
- Cluster Sampling: The population is divided into clusters, and a random sample of clusters is selected.
Key Concepts in Sampling
Sample Size and Population Size
- Sample Size (
): The number of individuals or items selected from the population. A larger typically leads to more accurate estimates. - Population Size (
): The total number of individuals or items in the population.
): The total number of individuals or items in the population.Sampling Error
Sampling error is the difference between the sample statistic and the population parameter. It decreases as 
increases:
![\[ \text{Sampling Error} = \bar{x} - \mu \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-76455ff3d0051f18d9f77563d842b731_l3.png "Rendered by QuickLaTeX.com")
where 
is the sample mean and 
is the population mean.
What Is Hypothesis Testing?
Hypothesis testing is a statistical method to make inferences about population parameters based on sample data. Key steps:
- Formulate Hypotheses: Null (
) and alternative (
). - Select Significance Level (): Typically 0.05.
- Calculate Test Statistic (e.g.,
or 
). - Make a Decision: Reject or fail to reject .
).
): Typically 0.05.
or 
).Null and Alternative Hypotheses
- Null Hypothesis (): No effect/difference (e.g.,
). - Alternative Hypothesis (): Contradicts (e.g.,
).
).
).Example Hypothesis
Test if the average height of students is 170 cm:
Significance Level and p-Value
Significance Level ()
Probability of Type I error (rejecting 
when true). Common values:
![\[ \alpha = 0.05, 0.01, \text{or } 0.10 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-26f09103ff2ed55d7ffb6a5346a8b0f6_l3.png "Rendered by QuickLaTeX.com")
p-Value
Probability of observing results as extreme as the sample data, assuming is true. Reject if:
![\[ p\text{-value} < \alpha \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-26f060349e7be13c79ee0b7b84105a43_l3.png "Rendered by QuickLaTeX.com")
Types of Errors
- Type I Error: Rejecting when true ().
- Type II Error (
): Failing to reject when false.
): Failing to reject Example Problem
Test if average student weight is 60 kg (
):
- Sample:
, 
, 
- ,
, 
, 
, 
Solution:
- Calculate -statistic:
![\[ z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} = \frac{62 - 60}{8/\sqrt{50}} = 1.77 \]](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMjIiIGhlaWdodD0iNDMiIHZpZXdCb3g9IjAgMCAyMjIgNDMiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+ "Rendered by QuickLaTeX.com")
- Critical value for
(two-tailed): 
. - Decision: Since
, fail to reject .
![\[ z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} = \frac{62 - 60}{8/\sqrt{50}} = 1.77 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-6c9e5100756fec854465120b37466107_l3.png "Rendered by QuickLaTeX.com")
.
, fail to reject Common Mistakes in Hypothesis Testing
- Not Defining Hypotheses Clearly: Make sure to clearly define both the null and alternative hypotheses before conducting the test.
- Confusing p-Value with Significance Level: The p-value is the probability of observing the test statistic under the null hypothesis, not the significance level.
- Misinterpreting Type I and Type II Errors: Be aware of the consequences of making a Type I or Type II error, and adjust the significance level accordingly.
Practice Questions
- A sample of 100 students has an average height of 160 cm. Test the hypothesis that the average height of students in the school is 165 cm at the 5% significance level.
- A new drug is tested on 200 patients. The null hypothesis is that the drug has no effect. The sample mean recovery time is 12 days with a standard deviation of 4 days. Perform a hypothesis test at the 1% significance level.
- In a factory, a machine is tested for accuracy. The null hypothesis is that the machine is accurate to within 0.5 mm. The sample measurement shows a mean of 0.8 mm with a standard deviation of 0.2 mm. Perform a hypothesis test at the 10% significance level.
- Test if average height is 165 cm (
, , ). - Drug test:
, , , , 
.
, 
, 
, 
, 
, 
, 
.Skinat Tuition | Where Expert Tutoring Meets Proven Results.