Question Find the degrees of freedom, critical values $\chi_{L}^{2}$ and $\chi_{R}^{2}$ , and $\sigma$ confidence interval for a normal distribution sample with $n=149, s=1.95$ (1000 cells/ $\mu \mathrm{L}$ ) at 99% confidence.

Studdy Solution

STEP 1

Assumptions
1. The sample size is $n = 149$ .
2. The sample standard deviation is $s = 1.95$ .
3. The confidence level is 99%.
4. The population is normally distributed.
5. The degrees of freedom $df = n - 1$ .
6. The critical values $\chi_{L}^{2}$ and $\chi_{R}^{2}$ are to be found corresponding to the 99% confidence interval for a chi-square distribution with $df$ degrees of freedom.

STEP 2

First, we need to find the degrees of freedom for the chi-square distribution.
$df = n - 1$

STEP 3

Now, plug in the given value for the sample size $n$ to calculate the degrees of freedom.
$df = 149 - 1$

STEP 4

Calculate the degrees of freedom.
$df = 148$

STEP 5

Next, we need to find the critical values $\chi_{L}^{2}$ and $\chi_{R}^{2}$ for the chi-square distribution at the 99% confidence level. These values are typically found using a chi-square distribution table or statistical software.

STEP 6

The critical value $\chi_{L}^{2}$ corresponds to the lower tail of the chi-square distribution for the given degrees of freedom and confidence level. This value is associated with the lower 0.5% of the distribution since the confidence level is 99%.

STEP 7

The critical value $\chi_{R}^{2}$ corresponds to the upper tail of the chi-square distribution for the given degrees of freedom and confidence level. This value is associated with the upper 0.5% of the distribution since the confidence level is 99%.

STEP 8

Using a chi-square distribution table or statistical software, find the critical values $\chi_{L}^{2}$ and $\chi_{R}^{2}$ for $df = 148$ at the 99% confidence level.

STEP 9

Once the critical values are found, we can calculate the confidence interval estimate of the population standard deviation $\sigma$ using the formula:
$\sigma = s \sqrt{\frac{df}{\chi^2}}$
where $\chi^2$ is either the critical value $\chi_{L}^{2}$ or $\chi_{R}^{2}$ , and $s$ is the sample standard deviation.

STEP 10

First, calculate the lower limit of the confidence interval for $\sigma$ using $\chi_{L}^{2}$ .
$\sigma_{lower} = s \sqrt{\frac{df}{\chi_{L}^{2}}}$

STEP 11

Next, calculate the upper limit of the confidence interval for $\sigma$ using $\chi_{R}^{2}$ .
$\sigma_{upper} = s \sqrt{\frac{df}{\chi_{R}^{2}}}$

STEP 12

After calculating both the lower and upper limits, we will have the 99% confidence interval estimate for the population standard deviation $\sigma$ .
The degrees of freedom is 148, and the critical values $\chi_{L}^{2}$ and $\chi_{R}^{2}$ , as well as the confidence interval estimate of $\sigma$ , will be provided once the critical values are obtained from the chi-square distribution table or statistical software. Since I cannot access such a table or software, I cannot provide the exact critical values or the confidence interval. However, the steps outlined above describe the process to obtain these values.

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Question Find the degrees of freedom, critical values $\chi_{L}^{2}$ and $\chi_{R}^{2}$ , and $\sigma$ confidence interval for a normal distribution sample with $n=149, s=1.95$ (1000 cells/ $\mu \mathrm{L}$ ) at 99% confidence.

Studdy Solution

STEP 1

STEP 2

First, we need to find the degrees of freedom for the chi-square distribution.
$df = n - 1$

STEP 3

Now, plug in the given value for the sample size $n$ to calculate the degrees of freedom.
$df = 149 - 1$

STEP 4

Calculate the degrees of freedom.
$df = 148$

STEP 5

Next, we need to find the critical values $\chi_{L}^{2}$ and $\chi_{R}^{2}$ for the chi-square distribution at the 99% confidence level. These values are typically found using a chi-square distribution table or statistical software.

STEP 6

The critical value $\chi_{L}^{2}$ corresponds to the lower tail of the chi-square distribution for the given degrees of freedom and confidence level. This value is associated with the lower 0.5% of the distribution since the confidence level is 99%.

STEP 7

The critical value $\chi_{R}^{2}$ corresponds to the upper tail of the chi-square distribution for the given degrees of freedom and confidence level. This value is associated with the upper 0.5% of the distribution since the confidence level is 99%.

STEP 8

Using a chi-square distribution table or statistical software, find the critical values $\chi_{L}^{2}$ and $\chi_{R}^{2}$ for $df = 148$ at the 99% confidence level.

STEP 9

STEP 10

First, calculate the lower limit of the confidence interval for $\sigma$ using $\chi_{L}^{2}$ .
$\sigma_{lower} = s \sqrt{\frac{df}{\chi_{L}^{2}}}$

STEP 11

Next, calculate the upper limit of the confidence interval for $\sigma$ using $\chi_{R}^{2}$ .
$\sigma_{upper} = s \sqrt{\frac{df}{\chi_{R}^{2}}}$

STEP 12

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Question Find the degrees of freedom, critical values χL2\chi_{L}^{2}χL2​ and χR2\chi_{R}^{2}χR2​, and σ\sigmaσ confidence interval for a normal distribution sample with n=149,s=1.95n=149, s=1.95n=149,s=1.95 (1000 cells/μL\mu \mathrm{L}μL) at 99% confidence.

Studdy Solution

STEP 1

STEP 2

First, we need to find the degrees of freedom for the chi-square distribution.df=n−1df = n - 1df=n−1

STEP 3

Now, plug in the given value for the sample size nnn to calculate the degrees of freedom.df=149−1df = 149 - 1df=149−1

STEP 4

Calculate the degrees of freedom.df=148df = 148df=148

STEP 5

Next, we need to find the critical values χL2\chi_{L}^{2}χL2​ and χR2\chi_{R}^{2}χR2​ for the chi-square distribution at the 99% confidence level. These values are typically found using a chi-square distribution table or statistical software.

STEP 6

The critical value χL2\chi_{L}^{2}χL2​ corresponds to the lower tail of the chi-square distribution for the given degrees of freedom and confidence level. This value is associated with the lower 0.5% of the distribution since the confidence level is 99%.

STEP 7

The critical value χR2\chi_{R}^{2}χR2​ corresponds to the upper tail of the chi-square distribution for the given degrees of freedom and confidence level. This value is associated with the upper 0.5% of the distribution since the confidence level is 99%.

STEP 8

Using a chi-square distribution table or statistical software, find the critical values χL2\chi_{L}^{2}χL2​ and χR2\chi_{R}^{2}χR2​ for df=148df = 148df=148 at the 99% confidence level.

STEP 9

STEP 10

First, calculate the lower limit of the confidence interval for σ\sigmaσ using χL2\chi_{L}^{2}χL2​.σlower=sdfχL2\sigma_{lower} = s \sqrt{\frac{df}{\chi_{L}^{2}}}σlower​=sχL2​df​​

STEP 11

Next, calculate the upper limit of the confidence interval for σ\sigmaσ using χR2\chi_{R}^{2}χR2​.σupper=sdfχR2\sigma_{upper} = s \sqrt{\frac{df}{\chi_{R}^{2}}}σupper​=sχR2​df​​

STEP 12

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Question Find the degrees of freedom, critical values $\chi_{L}^{2}$ and $\chi_{R}^{2}$ , and $\sigma$ confidence interval for a normal distribution sample with $n=149, s=1.95$ (1000 cells/ $\mu \mathrm{L}$ ) at 99% confidence.

First, we need to find the degrees of freedom for the chi-square distribution.
$df = n - 1$

Now, plug in the given value for the sample size $n$ to calculate the degrees of freedom.
$df = 149 - 1$

Calculate the degrees of freedom.
$df = 148$

Next, we need to find the critical values $\chi_{L}^{2}$ and $\chi_{R}^{2}$ for the chi-square distribution at the 99% confidence level. These values are typically found using a chi-square distribution table or statistical software.

The critical value $\chi_{L}^{2}$ corresponds to the lower tail of the chi-square distribution for the given degrees of freedom and confidence level. This value is associated with the lower 0.5% of the distribution since the confidence level is 99%.

The critical value $\chi_{R}^{2}$ corresponds to the upper tail of the chi-square distribution for the given degrees of freedom and confidence level. This value is associated with the upper 0.5% of the distribution since the confidence level is 99%.

Using a chi-square distribution table or statistical software, find the critical values $\chi_{L}^{2}$ and $\chi_{R}^{2}$ for $df = 148$ at the 99% confidence level.

First, calculate the lower limit of the confidence interval for $\sigma$ using $\chi_{L}^{2}$ .
$\sigma_{lower} = s \sqrt{\frac{df}{\chi_{L}^{2}}}$

Next, calculate the upper limit of the confidence interval for $\sigma$ using $\chi_{R}^{2}$ .
$\sigma_{upper} = s \sqrt{\frac{df}{\chi_{R}^{2}}}$