Question
The following TI-84 Plus calculator display presents a 95% confidence interval for the difference between two means. The sample sizes are nv1=7 and nv2 = 10.
2-SampTInt
(20.904, 74.134)
df = 12.28537157
xv1= 150.375
xv2 = 102.856
Svx1 = 25.724
Svx2 = 23.548
nv1 = 7
nv2 = 10
The point estimate of uv1 - uv2 = ?
2-SampTInt
(20.904, 74.134)
df = 12.28537157
xv1= 150.375
xv2 = 102.856
Svx1 = 25.724
Svx2 = 23.548
nv1 = 7
nv2 = 10
The point estimate of uv1 - uv2 = ?
Answers
Answer
The number of degrees of freedom the calculator used is ?
Answer
We are 95% confident that the difference between the means is between ? and ?
Answered by
GPT-4o mini
To find the point estimate of the difference between the two population means \( \mu_1 - \mu_2 \), we can use the sample means \( x_{v1} \) and \( x_{v2} \) provided for the two groups.
The point estimate for the difference is calculated as:
\[
\hat{\mu}_1 - \hat{\mu}_2 = x_{v1} - x_{v2}
\]
Given:
- \( x_{v1} = 150.375 \)
- \( x_{v2} = 102.856 \)
Now, substituting the values in:
\[
\hat{\mu}_1 - \hat{\mu}_2 = 150.375 - 102.856
\]
Calculating that gives:
\[
\hat{\mu}_1 - \hat{\mu}_2 = 47.519
\]
Thus, the point estimate of \( \mu_1 - \mu_2 \) is approximately **47.519**.
The point estimate for the difference is calculated as:
\[
\hat{\mu}_1 - \hat{\mu}_2 = x_{v1} - x_{v2}
\]
Given:
- \( x_{v1} = 150.375 \)
- \( x_{v2} = 102.856 \)
Now, substituting the values in:
\[
\hat{\mu}_1 - \hat{\mu}_2 = 150.375 - 102.856
\]
Calculating that gives:
\[
\hat{\mu}_1 - \hat{\mu}_2 = 47.519
\]
Thus, the point estimate of \( \mu_1 - \mu_2 \) is approximately **47.519**.
Answered by
GPT-4o mini
The number of degrees of freedom (df) used in the context of a two-sample t-test for the difference between means is typically calculated using a formula that combines the sample sizes and standard deviations of the two samples.
In your case, the TI-84 Plus calculator has provided a value for degrees of freedom:
\[
df = 12.28537157
\]
This is the number of degrees of freedom that the calculator used for the confidence interval. In general, for two-sample t-tests with potentially unequal variances, the degrees of freedom can be computed using the following formula (Welch's approximation):
\[
df = \frac{(S_1^2/n_1 + S_2^2/n_2)^2}{\frac{(S_1^2/n_1)^2}{n_1 - 1} + \frac{(S_2^2/n_2)^2}{n_2 - 1}}
\]
Where:
- \( S_1 \) and \( S_2 \) are the sample standard deviations of the first and second samples, respectively.
- \( n_1 \) and \( n_2 \) are the sample sizes for the first and second samples.
However, since the calculator's display already gives \( df = 12.28537157 \), that confirms the value used for your analysis.
So, the number of degrees of freedom used by the calculator is approximately **12.29** (rounded to two decimal places).
In your case, the TI-84 Plus calculator has provided a value for degrees of freedom:
\[
df = 12.28537157
\]
This is the number of degrees of freedom that the calculator used for the confidence interval. In general, for two-sample t-tests with potentially unequal variances, the degrees of freedom can be computed using the following formula (Welch's approximation):
\[
df = \frac{(S_1^2/n_1 + S_2^2/n_2)^2}{\frac{(S_1^2/n_1)^2}{n_1 - 1} + \frac{(S_2^2/n_2)^2}{n_2 - 1}}
\]
Where:
- \( S_1 \) and \( S_2 \) are the sample standard deviations of the first and second samples, respectively.
- \( n_1 \) and \( n_2 \) are the sample sizes for the first and second samples.
However, since the calculator's display already gives \( df = 12.28537157 \), that confirms the value used for your analysis.
So, the number of degrees of freedom used by the calculator is approximately **12.29** (rounded to two decimal places).
Answered by
GPT-4o mini
The 95% confidence interval for the difference between the means, as given in your original data from the TI-84 Plus calculator, is provided in the following format:
\[
(20.904, 74.134)
\]
This means that we are 95% confident that the true difference between the means ( \( \mu_1 - \mu_2 \) ) lies between:
- **Lower limit:** 20.904
- **Upper limit:** 74.134
So, we can say:
**We are 95% confident that the difference between the means is between 20.904 and 74.134.**
\[
(20.904, 74.134)
\]
This means that we are 95% confident that the true difference between the means ( \( \mu_1 - \mu_2 \) ) lies between:
- **Lower limit:** 20.904
- **Upper limit:** 74.134
So, we can say:
**We are 95% confident that the difference between the means is between 20.904 and 74.134.**
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