Answer :
To test the claim that the population mean of the "Tike" ratings is less than 6.00, let's go through the following steps:
### Step 1: State the Hypotheses
First, we need to establish our null and alternative hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The population mean is 6.00. Mathematically, it is [tex]\(H_0: \mu = 6.00\)[/tex].
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The population mean is less than 6.00. Mathematically, it is [tex]\(H_1: \mu < 6.00\)[/tex].
This matches with option C:
[tex]\[
H_0: \mu = 6.00 \quad \text{and} \quad H_1: \mu < 6.00
\][/tex]
### Step 2: Calculate the Test Statistic
We use a t-test for the mean because the sample standard deviation is given and the sample size is finite. The formula for the t-test statistic is:
[tex]\[
t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}
\][/tex]
Where:
- [tex]\(\bar{x} = 5.86\)[/tex] (Sample mean)
- [tex]\(\mu_0 = 6.00\)[/tex] (Hypothesized population mean)
- [tex]\(s = 2.01\)[/tex] (Sample standard deviation)
- [tex]\(n = 190\)[/tex] (Sample size)
The calculated t-statistic is approximately [tex]\(-0.96\)[/tex].
### Step 3: Determine the P-value
Using the t-distribution with [tex]\(n - 1\)[/tex] degrees of freedom (189 degrees of freedom in this case), we find the p-value associated with the calculated t-statistic. The p-value is approximately [tex]\(0.169\)[/tex].
### Step 4: Make a Decision
We compare the p-value to the significance level [tex]\(\alpha = 0.10\)[/tex].
- If the p-value is less than [tex]\(\alpha\)[/tex], we reject the null hypothesis.
- If the p-value is greater than or equal to [tex]\(\alpha\)[/tex], we fail to reject the null hypothesis.
Since [tex]\(0.169 > 0.10\)[/tex], we fail to reject [tex]\(H_0\)[/tex].
### Conclusion
Based on the analysis, there is not enough evidence at the 0.10 significance level to reject the null hypothesis. Therefore, we do not have sufficient evidence to support the claim that the population mean of the "Tike" ratings is less than 6.00.
### Step 1: State the Hypotheses
First, we need to establish our null and alternative hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The population mean is 6.00. Mathematically, it is [tex]\(H_0: \mu = 6.00\)[/tex].
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The population mean is less than 6.00. Mathematically, it is [tex]\(H_1: \mu < 6.00\)[/tex].
This matches with option C:
[tex]\[
H_0: \mu = 6.00 \quad \text{and} \quad H_1: \mu < 6.00
\][/tex]
### Step 2: Calculate the Test Statistic
We use a t-test for the mean because the sample standard deviation is given and the sample size is finite. The formula for the t-test statistic is:
[tex]\[
t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}
\][/tex]
Where:
- [tex]\(\bar{x} = 5.86\)[/tex] (Sample mean)
- [tex]\(\mu_0 = 6.00\)[/tex] (Hypothesized population mean)
- [tex]\(s = 2.01\)[/tex] (Sample standard deviation)
- [tex]\(n = 190\)[/tex] (Sample size)
The calculated t-statistic is approximately [tex]\(-0.96\)[/tex].
### Step 3: Determine the P-value
Using the t-distribution with [tex]\(n - 1\)[/tex] degrees of freedom (189 degrees of freedom in this case), we find the p-value associated with the calculated t-statistic. The p-value is approximately [tex]\(0.169\)[/tex].
### Step 4: Make a Decision
We compare the p-value to the significance level [tex]\(\alpha = 0.10\)[/tex].
- If the p-value is less than [tex]\(\alpha\)[/tex], we reject the null hypothesis.
- If the p-value is greater than or equal to [tex]\(\alpha\)[/tex], we fail to reject the null hypothesis.
Since [tex]\(0.169 > 0.10\)[/tex], we fail to reject [tex]\(H_0\)[/tex].
### Conclusion
Based on the analysis, there is not enough evidence at the 0.10 significance level to reject the null hypothesis. Therefore, we do not have sufficient evidence to support the claim that the population mean of the "Tike" ratings is less than 6.00.
