A sample of 25 measurements of the diameter of a sphere gave a mean of 47.5

cm and a standard deviation 8.4 cm. Does this information claim the fact that 

the mean of the population is 42.1 cm.

To assess whether the sample data supports the claim that the population mean diameter of the sphere is 42.1 cm, we can conduct a **hypothesis test**. Specifically, we can use a **one-sample t-test**, as the sample size is relatively small (n = 25), and we assume that the population follows a normal distribution.

### Hypothesis Setup:
- **Null Hypothesis (H₀)**: The population mean diameter is 42.1 cm.
  \[
  H_0: \mu = 42.1
  \]
- **Alternative Hypothesis (H₁)**: The population mean diameter is not 42.1 cm.
  \[
  H_1: \mu \neq 42.1
  \]

This is a **two-tailed test**, as we are interested in whether the mean is significantly different from 42.1 cm (either higher or lower).

### Given Data:
- Sample mean \( \bar{x} = 47.5 \, \text{cm} \)
- Population mean (claimed) \( \mu = 42.1 \, \text{cm} \)
- Sample standard deviation \( s = 8.4 \, \text{cm} \)
- Sample size \( n = 25 \)

### Test Statistic Formula:
The t-test statistic is calculated as:
\[
t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}
\]

### Step-by-step Calculation:

1. Calculate the **standard error** (SE):
\[
SE = \frac{s}{\sqrt{n}} = \frac{8.4}{\sqrt{25}} = \frac{8.4}{5} = 1.68
\]

2. Calculate the **t-statistic**:
\[
t = \frac{47.5 - 42.1}{1.68} = \frac{5.4}{1.68} \approx 3.21
\]

### Determine Critical Value:
For a two-tailed test with a significance level \( \alpha = 0.05 \) and degrees of freedom \( df = n - 1 = 24 \), we can find the critical t-value from a t-distribution table or a calculator.

At \( df = 24 \), the critical t-value for \( \alpha = 0.05 \) (two-tailed) is approximately **±2.064**.

### Conclusion:
- The calculated t-statistic is **3.21**, which is greater than the critical t-value of **2.064**.
- Therefore, we **reject the null hypothesis** at the 5% significance level.

### Interpretation:
There is sufficient evidence to suggest that the mean diameter of the population is significantly different from 42.1 cm. The sample data does **not support** the claim that the population mean is 42.1 cm.

---

Would you like to dive deeper into any specific part of this solution? Here are some related questions:

1. How do we interpret the significance level (α) in hypothesis testing?
2. What assumptions are needed for a t-test to be valid?
3. How does increasing the sample size affect the t-test result?
4. Could we use a Z-test in this scenario instead of a t-test?
5. How can we compute the p-value for the t-test?

**Tip**: In hypothesis testing, the smaller the p-value, the stronger the evidence against the null hypothesis.

A sample of 25 measurements of the diameter of a sphere gave a mean of 47.5 cm and a standard deviation of 8.4 cm. Does this information claim the fact that the mean of the population is 42.1 cm?

This example demonstrates how to use a one-sample t-test to evaluate a claim about the population mean based on a sample of 25 measurements of a sphere's diameter. The sample had a mean of 47.5 cm and a standard deviation of 8.4 cm. Learn the hypothesis testing process with detailed calculations and interpretation of statistical significance.

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