Question

Determine the null and alternative hypothesis
P-value

Answer 1

(a) **Hypotheses:**

`\[ H_0: \mu = 25 \]`

`\[ H_1: \mu \\eq 25 \]`

(b) **P-value:**

`\[ P\text{-value} \approx 0.140 \]`

Let's go through the solution step by step.

(a) **Null and Alternative Hypotheses:**

The null hypothesis (H0) is a statement that there is no significant difference or effect. In this case, it is that the population mean (\(\mu\)) is equal to 25.

`\[ H_0: \mu = 25 \]`

The alternative hypothesis (H1) is a statement that there is a significant difference or effect. In this case, it is that the population mean (\(\mu\)) is different from 25.

`\[ H_1: \mu \\eq 25 \]`

(b) **Calculate the P-value:**

To calculate the P-value, you would typically use a t-test, given that the population standard deviation is not known. The formula for the t-test statistic is:

`\[ t = \frac{{\bar{x} - \mu}}{{s/√(n)}} \]`

where:

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`\(\bar{x}\)` is the sample mean,

-
`\(\mu\)` is the hypothesized population mean under the null hypothesis,

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`\(s\)` is the sample standard deviation, and

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`\(n\)` is the sample size.

Given:

-
`\(\bar{x} = 28.1\)`

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`\(s = 6.3\)`

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`\(n = 16\)`

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`\(\mu_0\)` (the hypothesized population mean under the null hypothesis) is 25

`\[ t = \frac{{28.1 - 25}}{{6.3/√(16)}} \]\[ t = \frac{{3.1}}{{6.3/4}} \]\[ t \approx 1.968 \]`

Now, you need to find the P-value associated with this t-statistic. Since the alternative hypothesis is two-sided (stating that \(\mu\) is different from 25), you would look for the probability that a t-statistic is greater than 1.968 or less than -1.968 with 15 degrees of freedom (n-1).

You can use statistical software, a t-table, or an online calculator to find this probability. For example, using a t-distribution table or software, you find that the P-value is approximately 0.070.

So,

`\[ P\text{-value} \approx 2 * 0.070 \] (because it's a two-sided test)\[ P\text{-value} \approx 0.140 \]`

(Round to three decimal places as needed.)

Therefore, the P-value is approximately 0.140.