To calculate the test statistic, we'll need some key values from the dataset, specifically the sample mean 
x
ˉ
x
ˉ
 , the sample standard deviation 
s
s, and the sample size 
n
=
142
n=142. The null hypothesis mean 
μ
0
μ 
0
​
  is given as 271.

Let's assume hypothetical values for the sample mean and sample standard deviation, as they would typically be provided in the Statdisk output. For example, suppose:

Sample mean 
x
ˉ
=
260
x
ˉ
 =260
Sample standard deviation 
s
=
30
s=30
Given these values, the test statistic 
t
t can be calculated as:

t
=
x
ˉ
−
μ
0
s
n
t= 
n
​
 
s
​
 
x
ˉ
 −μ 
0
​
 
​


To calculate the test statistic \( t \), we use the formula:

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

where:
- \( \bar{x} = 260 \) is the sample mean,
- \( \mu_0 = 271 \) is the null hypothesis mean,
- \( s = 30 \) is the sample standard deviation,
- \( n = 142 \) is the sample size.

### Step-by-step Calculation:

1. **Compute the Standard Error (SE):**

\[
\text{SE} = \frac{s}{\sqrt{n}} = \frac{30}{\sqrt{142}}
\]

2. **Calculate the Test Statistic \( t \):**

\[
t = \frac{260 - 271}{\text{SE}}
\]

Let's compute these values.The standard error (SE) is approximately \(2.52\). The test statistic \( t \) is approximately \(-4.37\).

### Interpretation:
- A negative \( t \)-value indicates that the sample mean (\( \bar{x} = 260 \)) is less than the null hypothesis mean (\( \mu_0 = 271 \)).
- The magnitude of the \( t \)-value suggests how far the sample mean is from the null hypothesis mean in terms of standard errors. In this case, a \( t \)-value of \(-4.37\) indicates it is quite far.

Would you like to know more about the implications of this test statistic, such as p-value calculation or hypothesis testing interpretation?

Here are 5 related questions that expand on this topic:
1. What is the p-value associated with this test statistic, and how do we interpret it?
2. How do we determine if this \( t \)-value is statistically significant?
3. What assumptions must be met to perform this \( t \)-test?
4. How does the sample size affect the standard error and test statistic?
5. What are the differences between a one-tailed and two-tailed \( t \)-test?

**Tip:** Always check whether the sample data follows a normal distribution, especially when dealing with small sample sizes for \( t \)-tests.

To calculate the test statistic, we'll need some key values from the dataset, specifically the sample mean x̄, the sample standard deviation s, and the sample size n = 142. The null hypothesis mean μ0 is given as 271. Let's assume hypothetical values for the sample mean and sample standard deviation, as they would typically be provided in the Statdisk output. For example, suppose: Sample mean x̄ = 260 Sample standard deviation s = 30 Given these values, the test statistic t can be calculated as: t = (x̄ − μ0) / (s / √n).

Learn how to calculate the t-test statistic with a sample mean of 260, a sample standard deviation of 30, and a sample size of 142. This problem involves key statistical concepts such as hypothesis testing, standard error calculation, and the t-distribution. Ideal for undergraduate-level statistics students.

Math Problem Statement

To calculate the test statistic, we'll need some key values from the dataset, specifically the sample mean x ˉ x ˉ , the sample standard deviation s s, and the sample size n

Sample mean x ˉ

260 x ˉ =260 Sample standard deviation s

t

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