The test marks for 29 students studying STSA 1624 produced a mean of 72,8% and a variance of 275,56%. If the distribution of the marks may be assumed to be approximately normally distributed, determine the upper limit, if a 99% confidence interval is set up for the population variance for the population of 1700 registered students
619.24%
147.41%
173.54%
151.32%
503.96%
To calculate the upper limit of a 99% confidence interval for the population variance, we will use the chi-square distribution.
The chi-square distribution has degrees of freedom equal to the sample size minus 1, which in this case is 29 - 1 = 28.
We need to find the value of chi-square that corresponds to a cumulative probability of 0.01 (1% significance level) with 28 degrees of freedom.
Using a chi-square table or calculator, we find the value of chi-square to be approximately 52.29.
The upper limit of the 99% confidence interval for the population variance can be calculated as:
upper limit = (n - 1) * sample variance / chi-square
Where n = total sample size and sample variance = variance of the sample.
In this case, n = 1700 (population size) and sample variance = 275.56%.
upper limit = (1700 - 1) * 275.56% / 52.29
= 1699 * 2.7556 / 52.29
= 89,823.44 / 52.29
≈ 1,718.25
Therefore, the upper limit for the 99% confidence interval for the population variance is approximately 1,718.25.
None of the provided options match this value.
To determine the upper limit for a 99% confidence interval for the population variance, we can use the chi-square distribution.
The chi-square distribution is used to calculate confidence intervals for the population variance, given the sample variance and sample size.
Step 1: Calculate the chi-square statistic
The chi-square statistic is calculated using the sample variance and the sample size.
Chi-square statistic = ((n - 1) * sample variance) / population variance
Given:
Sample variance = 275.56%
Sample size = 29
Chi-square statistic = ((29 - 1) * 275.56%) / population variance
Step 2: Determine the critical value
To calculate the upper limit for the confidence interval, we need to determine the critical value from the chi-square distribution table. Because we want a 99% confidence interval, we look up the critical value for the upper tail probability of 0.01 and the degrees of freedom of n-1 (29-1).
Let's assume the critical value is denoted by "c".
Step 3: Calculate the upper limit
The upper limit for the confidence interval is calculated by dividing the chi-square statistic by the critical value.
Upper limit = chi-square statistic / critical value
Now we can calculate the upper limit.
Upper limit = chi-square statistic / c
= ((29 - 1) * 275.56%) / c
Therefore, the correct answer is dependent on the value of "c" obtained from the chi-square distribution table. Without that value, we cannot determine the exact upper limit for the 99% confidence interval.
To determine the upper confidence limit for the population variance, we can use the chi-square distribution with (n-1) degrees of freedom.
1. We are given the sample size (n = 29), the sample mean (x̄ = 72.8%), and the sample variance (s^2 = 275.56%).
2. Calculate the chi-square statistic using the formula:
χ² = (n-1) * s^2 / σ^2
Here, n is the sample size, s^2 is the sample variance, and σ^2 is the population variance.
3. Substitute the given values into the formula:
χ² = (29-1) * 275.56% / σ^2
= 28 * 275.56% / σ^2
4. Find the critical value from the chi-square distribution table for a 99% confidence level with (n-1) degrees of freedom. In this case, we have (28) degrees of freedom.
5. Look up the critical value in the chi-square distribution table. The closest value to the calculated χ² will give us the upper limit of the confidence interval for the population variance.
Let's calculate the chi-square statistic and find the corresponding critical value:
χ² = 28 * 275.56% / σ^2
Now, we need to consult the chi-square distribution table to find the critical value for a 99% confidence level with 28 degrees of freedom.
By comparing the calculated χ² with the critical value from the table, we can determine the upper limit of the confidence interval for the population variance.
Unfortunately, I cannot look up values in tables as I am a text-based bot. You can refer to a chi-square distribution table or use statistical software to find the critical value. Once you find it, compare it to the calculated χ² to determine the upper limit.
Please note that the options you provided in your question seem to be expressed in percentage terms, which is incorrect for a variance. Variance is not expressed as a percentage. So, the correct answer may not be among the options you provided.