the mean produce of wheat from a sample of 10 fields comes to 200kg/acre and another sample of 150 fields gives amean of 220kg/acre.Assuming the standard deviation of the yield at 11kg for population,test if there is a significant difference between the means of the sample at alpha=0.001
To test if there is a significant difference between the means of the two samples, we can use a two-sample t-test. Here's how you can calculate it step-by-step:
Step 1: Define the null and alternative hypotheses:
- Null Hypothesis: There is no significant difference between the means of the two samples.
- Alternative Hypothesis: There is a significant difference between the means of the two samples.
Step 2: Calculate the test statistic using the formula:
t = (Mean1 - Mean2) / sqrt((s1^2/n1) + (s2^2/n2))
Where:
Mean1 = 200 kg/acre
Mean2 = 220 kg/acre
s1 = standard deviation of the 10-field sample = 11 kg/acre
n1 = number of fields in the 10-field sample = 10
s2 = standard deviation of the 150-field sample = 11 kg/acre
n2 = number of fields in the 150-field sample = 150
Step 3: Calculate the degrees of freedom (df) using the formula:
df = (s1^2/n1 + s2^2/n2)^2 / ((s1^2/n1)^2/(n1-1) + (s2^2/n2)^2/(n2-1))
Step 4: Determine the critical value based on the alpha level and degrees of freedom. Since alpha = 0.001 and the degrees of freedom are calculated in Step 3, you can use a t-distribution table or statistical software to find the critical value.
Step 5: Compare the absolute value of the test statistic to the critical value. If the absolute value of the test statistic is greater than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
That's how you can test if there is a significant difference between the means of the two samples at an alpha level of 0.001.
To test if there is a significant difference between the means of the two samples, we can use a two-sample t-test. Here's how to calculate it:
Step 1: State the null hypothesis (H0) and alternate hypothesis (H1):
- Null hypothesis (H0): There is no significant difference between the means of the two samples.
- Alternate hypothesis (H1): There is a significant difference between the means of the two samples.
Step 2: Determine the significance level (alpha):
- Given in the question as alpha = 0.001.
Step 3: Calculate the critical value (t_critical) for a two-sample t-test:
- Use a critical value table or a t-distribution calculator with degrees of freedom equal to n1 + n2 - 2 (where n1 is the sample size of the first sample and n2 is the sample size of the second sample).
- Since the critical value depends on the significance level, degrees of freedom, and whether the test is one-tailed or two-tailed, we need to determine these factors before calculating the critical value.
Step 4: Calculate the test statistic (t-value):
- The formula for calculating the t-value is:
t = (x1 - x2) / sqrt((s1^2 / n1) + (s2^2 / n2))
where x1 and x2 are the sample means, s1 and s2 are the standard deviations, and n1 and n2 are the sample sizes.
Step 5: Compare the test statistic with the critical value:
- If the absolute value of the calculated t-value is greater than the critical value, reject the null hypothesis. There is a significant difference between the means of the two samples.
- If the absolute value of the calculated t-value is less than or equal to the critical value, fail to reject the null hypothesis. There is not enough evidence to conclude a significant difference between the means of the two samples.
Let's calculate the t-value and compare it with the critical value:
Given:
- Sample 1 (n1 = 10): mean1 = 200kg/acre
- Sample 2 (n2 = 150): mean2 = 220kg/acre
- Population standard deviation (σ) = 11kg
Step 1: State the hypotheses:
- H0: There is no significant difference between the means of the two samples (mean1 = mean2)
- H1: There is a significant difference between the means of the two samples (mean1 ≠ mean2)
Step 2: Significance level (alpha) = 0.001
Step 3: Critical value (t_critical) for a two-sample t-test:
- With degrees of freedom equal to n1 + n2 - 2, which is 10 + 150 - 2 = 158, and alpha = 0.001, consult a t-distribution table or use a t-distribution calculator to find t_critical.
- Let's assume that t_critical is approximately ±3.400 (for a two-tailed test).
Step 4: Calculate the test statistic (t-value):
- t = (mean1 - mean2) / sqrt((s1^2 / n1) + (s2^2 / n2))
- Since we have the population standard deviation (σ), we can estimate the sample standard deviation (s) using the following formula:
s = σ / sqrt(n)
- Plugging in the values:
s1 = 11 / sqrt(10) ≈ 3.48
s2 = 11 / sqrt(150) ≈ 0.90
- Now, calculate the t-value:
t = (200 - 220) / sqrt((3.48^2 / 10) + (0.90^2 / 150))
≈ -4.37
Step 5: Compare the test statistic with the critical value:
- Since this is a two-tailed test, we need to compare the absolute value of the t-value with the absolute value of the critical value.
- |t| = 4.37 > |t_critical| = 3.400
- Since the calculated t-value is greater than the critical value, we reject the null hypothesis.
Conclusion:
Based on the significance level (alpha = 0.001) and the calculated t-value, we have enough evidence to reject the null hypothesis. Therefore, there is a significant difference between the means of the two samples.