Discussion Board 13 CI & Hyp Single Pop Std Dev & Var - & Hyp Two Independent Population Standard Deviation
- You would like to construct a confidence interval for the standard deviation for the time it takes to prepare a tax return. You randomly select 70 tax returns and get a sample standard deviation of 8.3 minutes. Construct a 90% confidence interval for the standard deviation for the time it takes to prepare a tax return.
(Round margin of error to 3 decimal places at the end) and (express the final answer with 3 decimal places- such as 7.123 to 15.246). Make sure to use values from your tables. Test statistics, confidence interval values and probabilities generated by your calculator should not be used. Chris OByrne 2024
- Last year (Tax Year 2022), the standard deviation of tax return refunds was $1,140. You would like to see if the standard deviation (variability) has increased this tax season (2023). You randomly select 35 tax returns with refunds and get a standard deviation of $1,275. Conduct a hypothesis test to see if the standard deviation (variability) has increased this tax season and use a level of significance of .05. Chris OByrne 2024
A. What is the test statistics value (correct number of decimal places depending on which test statistic)?
B. What is the probability of the test statistic for this specific problem? (take into consideration if this is 1-tail or 2-tail test)
C. What conclusion will you come to? (just need to state Accept Ho or Reject Ho and Accept Ha just put one of those 2 answers in the body of the discussion board, but you will have a more detailed solution (besides writing one of these statements you will write out in words what the conclusion means) on your worksheet that you will attach as a PDF). Use a level of significance of .05.
D. Using the critical value approach at what critical value or values will you start rejecting the null hypothesis? (make sure you are very careful and detailed with your answer, pay attention to signs and values)
- You would like to see if the standard deviation (variability) of the earnings of stocks for the NASDAQ is greater than the standard deviation of the earnings of stocks for the S&P 500 for this current year. Use the information below and do a hypothesis test to see if the standard deviation (variability) of the earnings of stocks for the NASDAQ is greater than the standard deviation of the earnings of stocks for the S&P 500 for this current year. Use a level of significance of .05.
NASDAQS&P 500
45.3Sample Standard Deviation 37.1
30Sample Size45
A. What is the test statistics value (correct number of decimal places depending on which test statistic)?
B. What is the probability of the test statistic for this specific problem? (take into consideration if this is 1-tail or 2-tail test)
C. What conclusion will you come to? (just need to state Accept Ho or Reject Ho and Accept Ha just put one of those 2 answers in the body of the discussion board, but you will have a more detailed solution (besides writing one of these statements you will write out in words what the conclusion means) on your worksheet that you will attach as a PDF). Use a level of significance of .05.
D. Using the critical value approach at what critical value will you start rejecting the null hypothesis?
Chris OByrne 2024
Use the appropriate table or tables from our lecture notes to answer all questions that require values or probabilities Chris OByrne 2024
To address the discussion board questions effectively, we will perform the necessary calculations for each part related to confidence intervals and hypothesis testing. Let's break down the tasks step by step.
1. Confidence Interval for the Standard Deviation
To construct a 90% confidence interval for the standard deviation, we use the chi-square distribution. The formula for the confidence interval for the standard deviation is given by:
[
\left( \sqrt{\frac{(n-1) s^2}{\chi^2_{\alpha/2, n-1}}}, \sqrt{\frac{(n-1) s^2}{\chi^2_{1-\alpha/2, n-1}}} \right)
]
Where:
- ( n = 70 ) (sample size)
- ( s = 8.3 ) (sample standard deviation)
- ( \alpha = 0.10 ) (for a 90% confidence interval)
- Degrees of freedom (( df = n - 1 = 69 ))
From the chi-square distribution table:
- For ( \chi^2_{0.05, 69} ) (upper critical value): approximately 98.428
- For ( \chi^2_{0.95, 69} ) (lower critical value): approximately 45.722
Now we can calculate the confidence interval:
[
\text{Lower limit} = \sqrt{\frac{(70 - 1)(8.3^2)}{98.428}} = \sqrt{\frac{69 \cdot 68.89}{98.428}} \approx \sqrt{48.448} \approx 6.964
]
[
\text{Upper limit} = \sqrt{\frac{(70 - 1)(8.3^2)}{45.722}} = \sqrt{\frac{69 \cdot 68.89}{45.722}} \approx \sqrt{104.093} \approx 10.200
]
Thus, the 90% confidence interval for the standard deviation is:
6.964 to 10.200
2. Hypothesis Test for Increased Standard Deviation in Tax Refunds
We want to test if the standard deviation has increased from $1,140 to $1,275 using a significance level of ( \alpha = 0.05 ).
Hypotheses:
- Null hypothesis (( H_0 )): ( \sigma^2 = 1140^2 )
- Alternative hypothesis (( H_a )): ( \sigma^2 > 1140^2 )
Calculating the Test Statistic:
The test statistic is calculated using the formula:
[
\chi^2 = \frac{(n-1)s^2}{\sigma_0^2}
]
Where:
- ( n = 35 )
- ( s = 1275 )
- ( \sigma_0 = 1140 )
Substituting values:
[
\chi^2 = \frac{(35 - 1)(1275^2)}{1140^2} = \frac{34 \cdot 1625625}{1299600} \approx 43.184
]
A. Test Statistic Value:
43.184
B. Probability of the Test Statistic:
Using the chi-square distribution with ( n - 1 = 34 ) degrees of freedom, we find the p-value corresponding to ( \chi^2 = 43.184 ). This is a one-tailed test.
From a chi-square table or calculator, we find that the p-value is very small (less than ( 0.0005 )), indicating strong evidence against ( H_0 ).
C. Conclusion:
Reject ( H_0 ) and Accept ( H_a )
This means that there is sufficient evidence to conclude that the standard deviation of tax return refunds has increased this tax season.
D. Critical Value Approach:
For a one-tailed test at ( \alpha = 0.05 ) and ( df = 34):
The critical value from the chi-square table is approximately 50.000.
3. Hypothesis Test for Earnings Variability of NASDAQ vs S&P 500
Hypotheses:
- Null hypothesis (( H_0 )): ( \sigma_{NASDAQ}^2 = \sigma_{S&P500}^2 )
- Alternative hypothesis (( H_a )): ( \sigma_{NASDAQ}^2 > \sigma_{S&P500}^2 )
Using the formula for two independent population variances:
[
F = \frac{s_1^2}{s_2^2}
]
Where:
- ( s_1 = 45.3, n_1 = 30 ) (NASDAQ)
- ( s_2 = 37.1, n_2 = 45 ) (S&P 500)
Calculating:
[
F = \frac{45.3^2}{37.1^2} = \frac{2055.09}{1374.41} \approx 1.494
]
A. Test Statistic Value:
1.494
B. Probability of the Test Statistic:
Using an F-distribution with ( df_1 = n_1 - 1 = 29 ) and ( df_2 = n_2 - 1 = 44), we find that the p-value associated with ( F = 1.494 ) is approximately 0.10, indicating not significant evidence against ( H_0 ).
C. Conclusion:
Accept ( H_0 )
This means that there is insufficient evidence to conclude that the standard deviation of NASDAQ earnings is greater than that of the S&P 500.
D. Critical Value Approach:
For a one-tailed test at ( \alpha = 0.05):
The critical value from the F-table with ( df_1=29, df_2=44) is approximately 1.811.
This comprehensive breakdown addresses each question and provides clarity on statistical tests related to standard deviations in various contexts, ensuring proper calculations and interpretations based on established statistical methods.