You wish to test the following claim (
H
a
) at a significance level of
α
=
0.01
.

H
o
:
μ
=
89.9

H
a
:
μ
<
89.9

You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size
n
=
18
with mean
M
=
83.9
and a standard deviation of
S
D
=
14.1
.

What is the p-value for this sample? (Report answer accurate to four decimal places.)
p-value

To find the p-value for this sample, we need to conduct a one-sample t-test since the population standard deviation is unknown and the sample size is small (\(n = 18\)).

### Step 1: State the hypotheses
- Null hypothesis (\(H_0\)): \(\mu = 89.9\)
- Alternative hypothesis (\(H_a\)): \(\mu < 89.9\) (This is a one-tailed test)

### Step 2: Calculate the t-statistic
The t-statistic is calculated using the following formula:

\[
t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}
\]

Where:
- \(\bar{x} = 83.9\) (sample mean)
- \(\mu_0 = 89.9\) (hypothesized population mean)
- \(s = 14.1\) (sample standard deviation)
- \(n = 18\) (sample size)

Substitute the values into the formula:

\[
t = \frac{83.9 - 89.9}{\frac{14.1}{\sqrt{18}}}
\]

\[
t = \frac{-6}{\frac{14.1}{\sqrt{18}}}
\]

\[
t \approx \frac{-6}{3.323} \approx -1.805
\]

### Step 3: Determine the degrees of freedom
The degrees of freedom (\(df\)) for this t-test is:

\[
df = n - 1 = 18 - 1 = 17
\]

### Step 4: Find the p-value
Using the t-distribution table or a calculator, we find the p-value associated with the t-statistic \(-1.805\) with 17 degrees of freedom.

\[
\text{p-value} \approx 0.0441
\]

### Conclusion
The p-value for this sample is approximately **0.0441** (to four decimal places).

Do you want details or have any questions?

---

1. What is the significance level and how does it relate to the p-value?
2. How does the sample size affect the t-statistic and p-value?
3. What are degrees of freedom and how are they calculated?
4. What does a one-tailed test mean in hypothesis testing?
5. How can you use the t-distribution table to find the p-value?

**Tip:** When the population standard deviation is unknown and the sample size is small, the t-distribution should be used instead of the normal distribution.

Learn how to calculate the p-value for a one-sample t-test with unknown population standard deviation. This problem involves testing a hypothesis at a significance level of 0.01 using sample data with specific mean and standard deviation values. Get detailed steps and calculations for determining the p-value.

Math Problem Statement

You wish to test the following claim ( H a ) at a significance level of α

H o : μ

You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n

18 with mean M

83.9 and a standard deviation of S D

Solution

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