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Answer :
To solve this problem, we need to test the claim that the population mean rating given by female speed daters to male dates is less than 6.00, using the provided sample data. We will follow these steps:
### Step 1: Set up the null and alternative hypotheses.
- Null Hypothesis ([tex]\(H_0\)[/tex]): The population mean rating is 6.00. Mathematically, it is stated as [tex]\(H_0: \mu = 6.00\)[/tex].
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The population mean rating is less than 6.00. This is stated as [tex]\(H_1: \mu < 6.00\)[/tex].
### Step 2: Identify the given sample data.
- Sample size ([tex]\(n\)[/tex]) = 195
- Sample mean ([tex]\(\overline{x}\)[/tex]) = 5.82
- Sample standard deviation ([tex]\(s\)[/tex]) = 2.08
### Step 3: Calculate the test statistic.
The test statistic for a hypothesis test about a mean (with an unknown population standard deviation) is calculated using the t-statistic formula:
[tex]\[
t = \frac{\overline{x} - \mu_0}{s / \sqrt{n}}
\][/tex]
Where:
- [tex]\(\overline{x}\)[/tex] is the sample mean.
- [tex]\(\mu_0\)[/tex] is the hypothesized population mean (6.00 in this case).
- [tex]\(s\)[/tex] is the sample standard deviation.
- [tex]\(n\)[/tex] is the sample size.
Using the given data:
- [tex]\(\overline{x} = 5.82\)[/tex]
- [tex]\(\mu_0 = 6.00\)[/tex]
- [tex]\(s = 2.08\)[/tex]
- [tex]\(n = 195\)[/tex]
The calculated t-statistic is approximately [tex]\(-1.21\)[/tex].
### Step 4: Determine the degrees of freedom and p-value.
The degrees of freedom ([tex]\(df\)[/tex]) for the t-test is calculated as [tex]\(n - 1\)[/tex], which is [tex]\(195 - 1 = 194\)[/tex].
Using the t-distribution, the p-value corresponds to the probability of observing a t-statistic as extreme as or more extreme than [tex]\(-1.21\)[/tex] under the null hypothesis.
The p-value obtained from this test is approximately 0.114.
### Step 5: Make a decision based on the p-value.
The significance level ([tex]\(\alpha\)[/tex]) given is 0.05. We compare the p-value to the significance level to decide whether to reject the null hypothesis:
- If the p-value < [tex]\(\alpha\)[/tex]: Reject the null hypothesis.
- If the p-value [tex]\(\geq\)[/tex] [tex]\(\alpha\)[/tex]: Fail to reject the null hypothesis.
In this case, the p-value (0.114) is greater than the significance level (0.05).
### Conclusion:
Since the p-value is greater than [tex]\(\alpha\)[/tex], we fail to reject the null hypothesis. There is not enough evidence to support the claim that the population mean of the "like" ratings is less than 6.00.
### Step 1: Set up the null and alternative hypotheses.
- Null Hypothesis ([tex]\(H_0\)[/tex]): The population mean rating is 6.00. Mathematically, it is stated as [tex]\(H_0: \mu = 6.00\)[/tex].
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The population mean rating is less than 6.00. This is stated as [tex]\(H_1: \mu < 6.00\)[/tex].
### Step 2: Identify the given sample data.
- Sample size ([tex]\(n\)[/tex]) = 195
- Sample mean ([tex]\(\overline{x}\)[/tex]) = 5.82
- Sample standard deviation ([tex]\(s\)[/tex]) = 2.08
### Step 3: Calculate the test statistic.
The test statistic for a hypothesis test about a mean (with an unknown population standard deviation) is calculated using the t-statistic formula:
[tex]\[
t = \frac{\overline{x} - \mu_0}{s / \sqrt{n}}
\][/tex]
Where:
- [tex]\(\overline{x}\)[/tex] is the sample mean.
- [tex]\(\mu_0\)[/tex] is the hypothesized population mean (6.00 in this case).
- [tex]\(s\)[/tex] is the sample standard deviation.
- [tex]\(n\)[/tex] is the sample size.
Using the given data:
- [tex]\(\overline{x} = 5.82\)[/tex]
- [tex]\(\mu_0 = 6.00\)[/tex]
- [tex]\(s = 2.08\)[/tex]
- [tex]\(n = 195\)[/tex]
The calculated t-statistic is approximately [tex]\(-1.21\)[/tex].
### Step 4: Determine the degrees of freedom and p-value.
The degrees of freedom ([tex]\(df\)[/tex]) for the t-test is calculated as [tex]\(n - 1\)[/tex], which is [tex]\(195 - 1 = 194\)[/tex].
Using the t-distribution, the p-value corresponds to the probability of observing a t-statistic as extreme as or more extreme than [tex]\(-1.21\)[/tex] under the null hypothesis.
The p-value obtained from this test is approximately 0.114.
### Step 5: Make a decision based on the p-value.
The significance level ([tex]\(\alpha\)[/tex]) given is 0.05. We compare the p-value to the significance level to decide whether to reject the null hypothesis:
- If the p-value < [tex]\(\alpha\)[/tex]: Reject the null hypothesis.
- If the p-value [tex]\(\geq\)[/tex] [tex]\(\alpha\)[/tex]: Fail to reject the null hypothesis.
In this case, the p-value (0.114) is greater than the significance level (0.05).
### Conclusion:
Since the p-value is greater than [tex]\(\alpha\)[/tex], we fail to reject the null hypothesis. There is not enough evidence to support the claim that the population mean of the "like" ratings is less than 6.00.
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