The ages of competitors in a grandmaster chess tournament is under dispute. The National Chess League (NCL) claims that the average age of a grandmaster is 55. A random sample of 50 competitors at a recent tournament resulted in a mean of 59,18 and a variance of 168,60. Test at a 1% level of significance whether the NCL's claim is true.
To test whether the NCL's claim is true, we will use a one-sample t-test.
Null Hypothesis (H0): The average age of grandmasters is 55.
Alternative Hypothesis (H1): The average age of grandmasters is not equal to 55.
We will use a significance level of 1%.
Given that the sample mean (x̄) is 59.18 and the sample variance (s^2) is 168.60, we can calculate the standard error (SE) using the formula:
SE = sqrt(s^2/n)
Where n is the sample size. In this case, n = 50.
SE = sqrt(168.60/50) = sqrt(3.372) = 1.836
Next, we need to calculate the t-statistic using the formula:
t = (x̄ - μ) / (SE)
Where μ is the mean under the null hypothesis. In this case, μ = 55.
t = (59.18 - 55) / 1.836 = 2.38
Finally, we compare the t-statistic to the critical value from the t-distribution table. Since we're conducting a two-tailed test, we need to divide the significance level (1%) by 2 to get 0.5%.
df = n - 1 = 50 - 1 = 49
Critical value = ±2.675 (from t-distribution table with 0.5% significance level and 49 degrees of freedom)
Since the calculated t-statistic (2.38) does not exceed the critical value (±2.675), we fail to reject the null hypothesis. There is not enough evidence to support the claim that the average age of grandmasters is not equal to 55 at a 1% level of significance.
To test whether the NCL's claim is true, we need to perform a hypothesis test. Here are the steps:
Step 1: State the Null and Alternative hypotheses.
Null hypothesis (H0): The average age of grandmasters is 55.
Alternative hypothesis (H1): The average age of grandmasters is not 55.
Step 2: Determine the level of significance (α).
In this case, the level of significance is given as 1%, which means α = 0.01.
Step 3: Choose the appropriate test statistic.
Since the sample size is relatively large (n = 50) and the population variance is unknown, we'll use a t-test.
Step 4: Calculate the test statistic.
The test statistic for a t-test is given by:
t = (x̄ - μ) / (s / √n)
Where:
x̄ is the sample mean (59.18),
μ is the population mean (55),
s is the sample standard deviation (square root of the sample variance, so √168.60),
n is the sample size (50).
Plugging in the values, we get:
t = (59.18 - 55) / (√168.60 / √50)
Step 5: Determine the critical region.
Since we have a two-tailed test (H1: μ ≠ 55), we need to split the significance level (α) evenly between the two tails. In this case, we'll use α/2 = 0.01 / 2 = 0.005 for each tail.
Step 6: Calculate the degrees of freedom.
Since we have a sample size of 50, the degrees of freedom (df) is n - 1 = 50 - 1 = 49.
Step 7: Find the critical value(s).
Using a t-table or a t-distribution calculator for a two-tailed test with a significance level of 0.005 and 49 degrees of freedom, we find the critical value to be approximately ±2.68.
Step 8: Draw a conclusion and make a decision.
If the test statistic (t) falls within the critical region (outside of ±2.68), we reject the null hypothesis. Otherwise, if the test statistic falls within the non-critical region, we fail to reject the null hypothesis.
Let's calculate the test statistic to make a decision.
To test the claim made by the National Chess League (NCL) at a 1% level of significance, we need to perform a hypothesis test.
Hypothesis Testing Steps:
1. State the null hypothesis (H0) and the alternative hypothesis (H1):
- Null Hypothesis (H0): The average age of grandmasters is 55.
- Alternative Hypothesis (H1): The average age of grandmasters is not equal to 55.
2. Choose the appropriate test statistic:
- Since we have a sample mean and sample variance, and the sample size is relatively large (n > 30), we can use a z-test. The test statistic is the z-score.
3. Determine the critical value:
- We want to test at a 1% level of significance, which corresponds to a two-tailed test.
- The critical value for a two-tailed test at a 1% level of significance is ±2.58 (obtained from a standard normal distribution table).
4. Calculate the test statistic:
- The formula to calculate the z-score is:
z = (sample mean - population mean) / (standard deviation / sqrt(sample size))
- In this case, the population mean is 55, the sample mean is 59.18, the sample size is 50, and the variance is 168.60. To get the standard deviation, we take the square root of the variance.
z = (59.18 - 55) / (sqrt(168.60) / sqrt(50))
5. Compare the test statistic with the critical value:
- If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
6. State the conclusion:
- If we reject the null hypothesis, we conclude that there is evidence to support the alternative hypothesis. If we fail to reject the null hypothesis, we do not have enough evidence to support the alternative hypothesis.
Now, let's calculate the test statistic:
z = (59.18 - 55) / (sqrt(168.60) / sqrt(50))
z = 4.18 / (12.99 / 7.07)
z = 4.18 / 1.84
z ≈ 2.27
Since the absolute value of the test statistic (2.27) is less than the critical value (±2.58), we fail to reject the null hypothesis.
Conclusion:
There is not enough evidence to support the NCL's claim that the average age of grandmasters is 55 at a 1% level of significance.