Sure, let's go through the hypothesis testing step-by-step.
### Step 1: Setting up the Hypotheses
Null Hypothesis (H₀): The null hypothesis is that the population mean (μ) is equal to 4.5 hours. This can be written as:
[tex]\[ H₀: \mu = 4.5 \][/tex]
Alternative Hypothesis (H₁): The alternative hypothesis is that the population mean (μ) is not equal to 4.5 hours. This can be written as:
[tex]\[ H₁: \mu \neq 4.5 \][/tex]
### Step 2: Gathering Data and Known Values
- Sample size (n): 15
- Sample mean ( [tex]\(\bar{x} \)[/tex] ): 4.75
- Population mean (μ): 4.5
- Sample standard deviation (s): 2.0
- Population standard deviation (σ): 1.4
- Significance level (alpha): 0.05
### Step 3: Standard Error and Z-Score Calculation
Since the population standard deviation is known, we use the Z-test formula for the z-score:
The standard error (SE) is calculated as follows:
[tex]\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{1.4}{\sqrt{15}} \][/tex]
Using the z-score formula:
[tex]\[ z = \frac{\bar{x} - \mu}{SE} = \frac{4.75 - 4.5}{\frac{1.4}{\sqrt{15}}} \][/tex]
### Step 4: Finding the p-value
The p-value is the probability that the observed data would occur if the null hypothesis were true. It is computed based on the z-score obtained in the previous step.
### Step 5: Decision Making
We compare the p-value with the significance level (α = 0.05):
- If the p-value is less than the significance level, we reject the null hypothesis.
- If the p-value is greater than or equal to the significance level, we fail to reject the null hypothesis.
Here are the actual results from these computations:
- Z-score: 0.6916041689656102
- P-value: 0.4891859428731906
- Decision: The p-value (0.4892) is greater than the significance level (0.05).
### Final Decision:
Because the p-value is greater than the significance level, we do not have sufficient evidence to reject the null hypothesis. Therefore, we fail to reject the null hypothesis [tex]\( H₀: \mu = 4.5 \)[/tex]. Based on the data, it is reasonable to conclude that the average time teenagers spend on the phone per week is not significantly different from 4.5 hours.
