the mean production of a wheat of a sample 100 plots is 200 kg with standard deviations of 10 kg , another sample of 150 plots given mean production of the 200 kg with standard deviations of 12 kg assuming the standard deviations of the 11 kg for the universe, find at 1% level of significance whether the two results are consistece

To determine whether the two sample results are consistent, we need to test the null hypothesis \( H_0 \) that the two samples are from populations with the same mean, against the alternative hypothesis \( H_1 \) that the means are different. Given that both samples have equal mean (200 kg), we can perform a **two-sample Z-test** to check the consistency of the two samples.

### Information given:

- Sample 1:
  - \( n_1 = 100 \) (number of plots)
  - \( \overline{X}_1 = 200 \) kg (mean)
  - \( s_1 = 10 \) kg (standard deviation)

- Sample 2:
  - \( n_2 = 150 \) (number of plots)
  - \( \overline{X}_2 = 200 \) kg (mean)
  - \( s_2 = 12 \) kg (standard deviation)

- Population standard deviation (for both samples): \( \sigma = 11 \) kg

- Significance level: \( \alpha = 1\% = 0.01 \)

### Step-by-Step Procedure:

1. **Null Hypothesis ( \( H_0 \) )**: The two samples have the same population mean.
   \[
   H_0 : \mu_1 = \mu_2
   \]

2. **Alternative Hypothesis ( \( H_1 \) )**: The two samples have different population means.
   \[
   H_1 : \mu_1 \neq \mu_2
   \]

3. **Standard Error of the Difference Between the Means:**

Since both samples have the same population standard deviation, we calculate the **standard error** (SE) of the difference between the two means using:
\[
SE = \sqrt{\left(\frac{\sigma^2}{n_1}\right) + \left(\frac{\sigma^2}{n_2}\right)}
\]
where \( \sigma = 11 \), \( n_1 = 100 \), and \( n_2 = 150 \).

4. **Test Statistic (Z-value):**

The Z-value is calculated as:
\[
Z = \frac{\overline{X}_1 - \overline{X}_2}{SE}
\]
Since both sample means are the same (\( \overline{X}_1 = \overline{X}_2 = 200 \)), the numerator will be zero, leading to:
\[
Z = 0
\]

5. **Critical Z-value at 1% Significance Level:**

For a two-tailed test at \( \alpha = 0.01 \), the critical Z-value is approximately:
\[
Z_{\alpha/2} = 2.58
\]
(You can obtain this value from standard Z-tables for a two-tailed test at the 1% significance level.)

### Conclusion:

Since the calculated Z-value (0) is less than the critical Z-value (2.58), we **fail to reject the null hypothesis** \( H_0 \). This means that there is not enough evidence at the 1% significance level to suggest that the two samples are from populations with different means.

Thus, the two results are **consistent** at the 1% level of significance.

---

Would you like more details or have any questions about the calculations?

Here are 5 related questions:
1. What is the difference between a Z-test and a T-test?
2. How is the standard error affected by sample size?
3. What does it mean to reject the null hypothesis in a statistical test?
4. How do you interpret the p-value in hypothesis testing?
5. What is the importance of the level of significance in statistical tests?

**Tip:** Always check if your test should be one-tailed or two-tailed based on the alternative hypothesis to avoid wrong conclusions.

The mean production of wheat from a sample of 100 plots is 200 kg with a standard deviation of 10 kg. Another sample of 150 plots gives a mean production of 200 kg with a standard deviation of 12 kg. Assuming the standard deviation for the universe is 11 kg, find at the 1% level of significance whether the two results are consistent.

Determine if the mean production of wheat from two samples (100 plots and 150 plots) is consistent using a two-sample Z-test. Both samples have a mean production of 200 kg. With a population standard deviation of 11 kg, we perform the test at the 1% level of significance.

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