Questions: We test if artificial sunlight during the winter months lowers one's depression. Without the light, a depression test has μ=8. With the light, our sample with N=41 produced X̄=6. The tcat =-1.83. 1. What are the hypotheses? 2. What is tot ? 3. What is the conclusion? 4. If N had been 50, would the results be significant?

We test if artificial sunlight during the winter months lowers one's depression. Without the light, a depression test has μ=8. With the light, our sample with N=41 produced X̄=6. The tcat =-1.83.

1. What are the hypotheses?
2. What is tot ?
3. What is the conclusion?
4. If N had been 50, would the results be significant?

Transcript text: We test if artificial sunlight during the winter months lowers one's depression. Without the light, a depression test has $\mu=8$. With the light, our sample with $N=41$ produced $\bar{X}=6$. The $t_{\text {cat }}=-1.83$. 1. What are the hypotheses? 2. What is $t_{\text {ot }}$ ? 3. What is the conclusion? 4. If $N$ had been 50, would the results be significant?

Solution

Solution Steps

Step 1: Calculate the Standard Error

The standard error $ SE $ is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{14.0}{\sqrt{25}} = 2.8 \]

Step 2: Calculate the Test Statistic

The test statistic $ t_{\text{test}} $ is calculated using the formula: \[ t_{\text{test}} = \frac{\bar{x} - \mu_0}{SE} = \frac{46 - 40}{2.8} = 2.1429 \]

Step 3: Calculate the P-value

For a two-tailed test, the P-value is calculated as: \[ P = 2 \times (1 - T(|z|)) = 0.0425 \]

Step 4: State the Hypotheses

For the second scenario, the hypotheses are: