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A data set about speed dating includes "Tike" ratings of male dates made by the female dates. The summary statistics are [tex]n=190, \bar{x}=5.86, s=2.01[/tex]. Use a 0.10 significance level to test the claim that the population mean of such ratings is less than 6.00. Assume that a simple random sample has been selected. Identify the null and alternative hypotheses, test statistic, P-value, and state the final conclusion that addresses the original claim.

**What are the null and alternative hypotheses?**

A. [tex]H_0: \mu=6.00[/tex], [tex]H_1: \mu\ \textless \ 6.00[/tex]

B. [tex]H_0: \mu=6.00[/tex], [tex]H_1: \mu\ \textgreater \ 6.00[/tex]

C. [tex]H_0: \mu=6.00[/tex], [tex]H_1: \mu\neq6.00[/tex]

D. [tex]\begin{aligned} H_0: \mu\ \textless \ 6.00 \\ H_1: \mu\ \textgreater \ 6.00\end{aligned}[/tex]

**Determine the test statistic.**

[tex]\square[/tex] (Round to two decimal places as needed.)

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.

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