Question 1180575: State the null hypothesis and the alternative hypothesis in (a) words and in
(b) symbols for each of the following.
A random sample of 200 students got a mean score of 62 with a standard deviation score of 5 in a knowledge test in mathematics. In the standardization, μ = 50 with σ = 10.
Sketch the graph of each null and alternative hypothesis (one-tailed test and two-tailed test)
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to state the null and alternative hypotheses, both in words and symbols, and how to sketch their graphs:
**Scenario:** A random sample of 200 students has a mean score of 62 and a standard deviation of 5 on a math knowledge test. The population mean is 50 with a standard deviation of 10.
Since the sample mean (62) is greater than the population mean (50), we can assume that the researchers are testing if the new method or intervention (if any) caused an improvement in the scores. We will consider both one-tailed (right-tailed) and two-tailed tests.
**(a) Hypotheses in Words and Symbols:**
**One-Tailed Test (Right-Tailed):**
* **Null Hypothesis (H0):** The mean score of students is equal to 50.
* **Alternative Hypothesis (H1):** The mean score of students is greater than 50.
* **H0: μ = 50**
* **H1: μ > 50**
**Two-Tailed Test:**
* **Null Hypothesis (H0):** The mean score of students is equal to 50.
* **Alternative Hypothesis (H1):** The mean score of students is *not* equal to 50.
* **H0: μ = 50**
* **H1: μ ≠ 50**
**(b) Graphs of Hypotheses:**
The graphs below represent the distribution of the sample means. The shaded areas represent the rejection regions.
**One-Tailed Test (Right-Tailed):**
```
H0: μ = 50 H1: μ > 50
--------------------- ---------------------
| | | / |
| | | / |
| / | | / |
| / | | / |
| / | | / |
| / | | / |
| / | | / |
| / | | / |
| / | |/ |
--------------------- ---------------------
50 x̄ 50 x̄
```
The rejection region is on the right side of the distribution.
**Two-Tailed Test:**
```
H0: μ = 50 H1: μ ≠ 50
--------------------- ---------------------
| / | | / \ |
| / | | / \ |
| / | | / \ |
| / | | / \ |
| / | | / \ |
| / | | / \ |
| / | |/ \ |
| / | | \ |
|/ | | \ |
--------------------- ---------------------
50 x̄ 50 x̄
```
The rejection regions are on both sides of the distribution.
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