A random sample of stock prices per share (in dollars) is shown. Find the 90% confidence interval for the variance and standard deviation for the prices. Assume the variable is normally distributed.
Prices not shown.
Same process as later post, except:
90% = mean ± 1.645 SD
Solution to the question
To find the 90% confidence interval for the variance and standard deviation of the stock prices, we need to follow these steps:
Step 1: Calculate the sample variance, denoted as S^2. This is done by taking the sum of the squared deviations from the mean and dividing it by one less than the sample size.
Step 2: Calculate the chi-square value associated with the confidence level. For a 90% confidence level, we need to find the chi-square value that corresponds to a 5% significance level. This is equivalent to (1-Confidence level)/2. Since we want a 90% confidence level, the significance level is 0.05.
Step 3: Determine the degrees of freedom (df). For a sample variance, the degrees of freedom are equal to the sample size minus one.
Step 4: Calculate the lower and upper limits for the variance. The lower limit is obtained by multiplying the sample variance by (df/chisquare_upper), and the upper limit is obtained by multiplying the sample variance by (df/chisquare_lower).
Step 5: Calculate the lower and upper limits for the standard deviation. They are obtained by taking the square root of the variance limits calculated in step 4.
Using these steps, we can find the 90% confidence interval for the variance and standard deviation of the stock prices.
To find the confidence interval for the variance and standard deviation, we need to follow these steps:
Step 1: Calculate the sample variance
- Calculate the mean of the sample.
- For each observation, subtract the mean and square the result.
- Sum up all the squared differences and divide by (n-1), where n is the sample size. This will give you the sample variance.
Step 2: Find the critical value
- Determine the degrees of freedom (df) for the sample, which is equal to (n - 1), where n is the sample size.
- Look up the critical value for a 90% confidence level and the determined degrees of freedom in the Chi-Square table.
Step 3: Calculate the confidence interval for the variance
- Multiply the sample variance by (df / upper critical value) to get the lower bound.
- Multiply the sample variance by (df / lower critical value) to get the upper bound.
Step 4: Calculate the confidence interval for the standard deviation
- Take the square root of the lower bound of the variance to get the lower bound of the standard deviation.
- Take the square root of the upper bound of the variance to get the upper bound of the standard deviation.
It's important to note that the sample needs to be random and representative of the population. Additionally, assuming the variable is normally distributed is a crucial condition for this approach.
If you provide the sample of stock prices per share, I can assist you further in finding the confidence interval for the variance and standard deviation.