In statistics, hypothesis testing is a common method used to determine if there is a significant difference between a sample and a known population. One of the most commonly used tests in hypothesis testing is the one-sample t-test and z-test. In this lesson, we will explore the differences between the two tests and how they are used in practice.

The one-sample t-test is a statistical test used to determine if a sample mean is significantly different from a known population mean. It is used when the population standard deviation is unknown and must be estimated from the sample data. The formula for the one-sample t-test is:

```
t = (x̄ - μ) / (s / √n)
```

where:

- t = the t-statistic
- x̄ = the sample mean
- μ = the known population mean
- s = the sample standard deviation
- n = the sample size

The t-statistic is used to calculate the p-value, which is the probability of obtaining a sample mean as extreme as the one observed, assuming the null hypothesis is true. If the p-value is less than the significance level (typically 0.05), we reject the null hypothesis and conclude that there is a significant difference between the sample mean and the population mean.

**Example 1:**

Suppose a company wants to know if the average time spent on their website per session is significantly different from the industry average of 5 minutes. They collect a random sample of 50 website sessions and find that the average time spent is 4.5 minutes with a standard deviation of 1.2 minutes. They can use the one-sample t-test to test the hypothesis that the company's website time is equal to the industry average.

The null hypothesis would be: H0: μ = 5 The alternative hypothesis would be: Ha: μ ≠ 5

Using the formula for the one-sample t-test, we get:

```
t = (4.5 - 5) / (1.2 / √50) = -2.08
```

With 49 degrees of freedom (n-1), and a significance level of 0.05, we find the critical t-value to be ±2.009. Since our calculated t-value (-2.08) falls outside the critical region, we reject the null hypothesis and conclude that the average time spent on the company's website is significantly different from the industry average.

The z-test is a statistical test used to determine if a sample mean is significantly different from a known population mean. It is used when the population standard deviation is known. The formula for the z-test is:

```
z = (x̄ - μ) / (σ / √n)
```

where:

- z = the z-statistic
- x̄ = the sample mean
- μ = the known population mean
- σ = the population standard deviation
- n = the sample size

The z-statistic is used to calculate the p-value, which is the probability of obtaining a sample mean as extreme as the one observed, assuming the null hypothesis is true. If the p-value is less than the significance level (typically 0.05), we reject the null hypothesis and conclude that there is a significant difference between the sample mean and the population mean.

Z-test

**Example 2:**

Suppose a company wants to know if the average weight of their product is significantly different from the target weight of 500 grams. They collect a random sample of 100 products and find that the average weight is 498 grams with a standard deviation of 10 grams. They know from previous production runs that the population standard deviation is 5 grams. They can use the z-test to test the hypothesis that the product weight is equal to the target weight.

The null hypothesis would be: H0: μ = 500 The alternative hypothesis would be: Ha: μ ≠ 500

Using the formula for the z-test, we get:

```
z = (498 - 500) / (5 / √100) = -4
```

The main difference between the one-sample t-test and z-test is in the assumptions made about the population standard deviation. The one-sample t-test assumes that the population standard deviation is unknown and must be estimated from the sample data, while the z-test assumes that the population standard deviation is known.

Another difference is in the distribution used to calculate the p-value. The one-sample t-test uses the t-distribution, which has heavier tails than the normal distribution used in the z-test. This means that the t-test is more conservative and requires a larger sample size to achieve the same level of power as the z-test.

Difference between One-sample t-test and z-test

The one-sample t-test and z-test are both useful statistical tests for hypothesis testing. The choice between the two tests depends on whether the population standard deviation is known or unknown. The one-sample t-test is more commonly used because the population standard deviation is often unknown. However, if the population standard deviation is known, the z-test is a more powerful test.

Here are some key takeaways from this lesson:

- Hypothesis testing is a common method used to determine if there is a significant difference between a sample and a known population.
- The one-sample t-test is used when the population standard deviation is unknown and must be estimated from the sample data, while the z-test is used when the population standard deviation is known.
- Both tests use the sample mean, population mean, sample size, and a calculated statistic (t or z) to determine the p-value and make conclusions about the null hypothesis.
- The t-test is more conservative and requires a larger sample size to achieve the same level of power as the z-test because it uses the t-distribution, which has heavier tails than the normal distribution used in the z-test.
- The choice between the two tests depends on whether the population standard deviation is known or unknown. The one-sample t-test is more commonly used because the population standard deviation is often unknown.

**1. What is the one-sample t-test used for?**

- To determine if a sample mean is significantly different from a known population mean when the population standard deviation is unknown.
- To determine if a sample mean is significantly different from a known population mean when the population standard deviation is known.
- To determine if a sample proportion is significantly different from a known population proportion.
- To determine if two samples are significantly different from each other.

**Answer**: **a.** To determine if a sample mean is significantly different from a known population mean when the population standard deviation is unknown.

**2. What is the z-test used for?**

- To determine if a sample mean is significantly different from a known population mean when the population standard deviation is unknown.
- To determine if a sample mean is significantly different from a known population mean when the population standard deviation is known.
- To determine if a sample proportion is significantly different from a known population proportion.
- To determine if two samples are significantly different from each other.

**Answer**: b. To determine if a sample mean is significantly different from a known population mean when the population standard deviation is known.

**3. What is the main difference between the one-sample t-test and z-test?**

- The distribution used to calculate the p-value.
- The assumptions made about the population standard deviation.
- The type of data being tested.
- The number of samples being compared.

**Answer**: b. The assumptions made about the population standard deviation.

**4. Which test is more commonly used and why?**

a. The one-sample t-test, because the population standard deviation is often unknown.

b. The z-test, because the population standard deviation is often unknown.

c. The one-sample t-test, because the population standard deviation is often known.

d. The z-test, because the population standard deviation is often known.

**Answer**: a. The one-sample t-test, because the population standard deviation is often unknown.

Module 7: Hypothesis Testing

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