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 can perform a one-sample t-test. The formula for the one-sample t-test is:

\[
t = \frac{\bar{X} - \mu}{\frac{s}{\sqrt{n}}}
\]

Where:
- \(\bar{X}\) is the sample mean (70.55 grams),
- \(\mu\) is the population mean (70 grams),
- \(s\) is the sample standard deviation (2.82 grams),
- \(n\) is the sample size (20).

### Step-by-step solution:

1. **Calculate the standard error (SE)**:
   \[
   SE = \frac{s}{\sqrt{n}} = \frac{2.82}{\sqrt{20}}
   \]

2. **Calculate the t-statistic**:
   \[
   t = \frac{70.55 - 70}{SE}
   \]

3. **Determine degrees of freedom (df)**: 
   \[
   df = n - 1 = 20 - 1 = 19
   \]

Finally, we compare the calculated t-value to the critical value from the t-distribution table for 19 degrees of freedom at a given significance level (e.g., 0.05). If the calculated t-value exceeds the critical value, the difference is significant. 

Let’s calculate this.The calculated t-statistic is approximately \( t = 0.87 \), with a standard error (SE) of 0.63. The degrees of freedom (df) are 19.

To determine if this t-statistic is significant, we need to compare it against a critical value from the t-distribution table for 19 degrees of freedom at a chosen significance level (e.g., \(\alpha = 0.05\)). If the absolute value of the t-statistic exceeds the critical value, the difference is significant.

Would you like me to look up the critical t-value or test at a specific significance level (e.g., 0.05 or 0.01)? 

---

Here are some related questions to deepen understanding:
1. What are the assumptions for performing a one-sample t-test?
2. How do you interpret the p-value in hypothesis testing?
3. What is the difference between a one-sample t-test and a two-sample t-test?
4. How would the result change if the sample size was larger?
5. Why is the t-distribution used instead of the normal distribution in this case?

**Tip:** In hypothesis testing, the null hypothesis assumes no difference between the sample mean and the population mean.

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.

Learn how to determine if the average weight of a sample of 20 mangoes differs significantly from the population mean of 70 grams using a one-sample t-test. With a sample mean of 70.55 grams and a standard deviation of 2.82 grams, this problem guides you through calculating the t-statistic and comparing it to critical values.

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