Statistics and Probability: Sampling and Hypothesis Testing

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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 ( $n$ ): The number of individuals or items selected from the population. A larger $n$ typically leads to more accurate estimates.
Population Size ( $N$ ): 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 $n$ increases:

$\text{Sampling Error} = \bar{x} - \mu$

where $\bar{x}$ is the sample mean and $\mu$ 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 ( $H_0$ ) and alternative ( $H_1$ ).
Select Significance Level ( $\alpha$ ): Typically 0.05.
Calculate Test Statistic (e.g., $z$ or $t$ ).
Make a Decision: Reject or fail to reject $H_0$ .

Null and Alternative Hypotheses

Null Hypothesis ( $H_0$ ): No effect/difference (e.g., $H_0: \mu = \mu_0$ ).
Alternative Hypothesis ( $H_1$ ): Contradicts $H_0$ (e.g., $H_1: \mu \neq \mu_0$ ).

Example Hypothesis

Test if the average height of students is 170 cm:

$H_0: \mu = 170$
$H_1: \mu \neq 170$

Significance Level and p-Value

Significance Level ( $\alpha$ )

Probability of Type I error (rejecting $H_0$ when true). Common values:

$\alpha = 0.05, 0.01, \text{or } 0.10$

p-Value

Probability of observing results as extreme as the sample data, assuming $H_0$ is true. Reject $H_0$ if:

$p\text{-value} < \alpha$

Types of Errors

Type I Error: Rejecting $H_0$ when true ( $\alpha$ ).
Type II Error ( $\beta$ ): Failing to reject $H_0$ when false.

Example Problem

Test if average student weight is 60 kg ( $\alpha = 0.05$ ):

Sample: $n = 50$ , $\bar{x} = 62$ , $\sigma = 8$
$H_0: \mu = 60$ , $H_1: \mu \neq 60$

Solution:

Calculate $z$ -statistic:
$z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} = \frac{62 - 60}{8/\sqrt{50}} = 1.77$
Critical value for $\alpha = 0.05$ (two-tailed): $\pm 1.96$ .
Decision: Since $1.77 < 1.96$ , fail to reject $H_0$ .

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 ( $n = 100$ , $\bar{x} = 160$ , $\alpha = 0.05$ ).
Drug test: $H_0: \text{no effect}$ , $\bar{x} = 12$ , $\sigma = 4$ , $n = 200$ , $\alpha = 0.01$ .

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