Determine whether the average weight of a sample of 20 mangoes is significantly different
from the population’s average weight of 70 grams. The sample mean weight is 70.55 grams,
and the sample standard deviation is 2.82 grams. Use one sample T-test.

To determine whether the average weight of the sample of 20 mangoes is significantly different from the population's average weight of 70 grams, we'll perform a one-sample T-test.

### Given Data:
- Population mean (\(\mu\)): 70 grams
- Sample mean (\(\bar{x}\)): 70.55 grams
- Sample standard deviation (\(s\)): 2.82 grams
- Sample size (\(n\)): 20

### Step 1: State the hypotheses

- **Null hypothesis (\(H_0\))**: The sample mean is equal to the population mean.
  \[
  H_0: \mu = 70 \text{ grams}
  \]
- **Alternative hypothesis (\(H_1\))**: The sample mean is different from the population mean.
  \[
  H_1: \mu \neq 70 \text{ grams}
  \]

### Step 2: Calculate the test statistic

The formula for the T-test statistic is:
\[
t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}
\]

Substituting the given values:
\[
t = \frac{70.55 - 70}{\frac{2.82}{\sqrt{20}}}
\]

Let's calculate the value:

\[
t = \frac{0.55}{\frac{2.82}{\sqrt{20}}} \approx \frac{0.55}{0.6308} \approx 0.8718
\]

### Step 3: Determine the degrees of freedom and critical value

- Degrees of freedom (\(df\)) = \(n - 1 = 20 - 1 = 19\)
- For a two-tailed test at the 0.05 significance level, the critical value of \(t\) for \(df = 19\) is approximately \(\pm 2.093\) (from the T-distribution table).

### Step 4: Compare the test statistic to the critical value

- If \(|t| > t_{\text{critical}}\), we reject the null hypothesis.
- Here, \(|t| = 0.8718\) and \(t_{\text{critical}} = 2.093\).

Since \(0.8718\) is less than \(2.093\), we fail to reject the null hypothesis.

### Conclusion:
There is no significant evidence to suggest that the average weight of the sample of mangoes is different from the population’s average weight of 70 grams.

Would you like further details on this test, or do you have any questions?

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**Relative Questions:**
1. How would the conclusion change if the sample mean was 72 grams?
2. What is the significance level in hypothesis testing, and how does it affect the conclusion?
3. Can you explain the concept of degrees of freedom in a T-test?
4. How does increasing the sample size impact the T-test results?
5. What are the assumptions required for conducting a one-sample T-test?

**Tip:** Always check the assumptions of normality and the sample size when performing a T-test, as these can affect the validity of the test results.

Learn how to perform a one-sample T-test to determine if the average weight of a sample of 20 mangoes is significantly different from the population's average weight of 70 grams. Includes step-by-step calculation using sample mean, sample standard deviation, and population mean. Suitable for undergraduate students studying statistical inference.

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