Problem 8 Finding Critical Values and Conf... [FREE SOLUTION] (2024)

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Chapter 7: Problem 8

Finding Critical Values and Confidence Intervals. In Exercises \(5-8,\) use thegiven information to find the number of degrees of freedom, the criticalvalues \(\mathcal{X}_{L}^{2}\) and \(\mathcal{X}_{R}^{2},\) and the confidenceinterval estimate of \(\boldsymbol{\sigma} .\) The samples are from Appendix\(\boldsymbol{B}\) and it is reasonable to assume that a simple random samplehas been selected from a population with a normal distribution. Heights of Men \(99 \%\) confidence; \(n=153, s=7.10 \mathrm{cm}\)

Short Answer

Expert verified

Degrees of Freedom: 152. Critical values: \(\mathcal{X}_{L}^{2} = 118.528\) and \(\mathcal{X}_{R}^{2} = 191.036\). Confidence interval for \(\sigma\): \(6.32\) to \(8.02\).

Step by step solution

02

Determine Critical Values

For a 99% confidence interval, the significance level \( \alpha \) is 0.01. The left and right critical values are found from the chi-square distribution table corresponding to \( df = 152 \) and \( \alpha/2 \) and \( 1 - \alpha/2 \) respectively. Thus: \( \alpha/2 = 0.005 \) and \( 1 - \alpha/2 = 0.995 \).From the chi-square distribution table, for \( df = 152 \): \[ \mathcal{X}_{L}^{2} = 118.528 \space (value \space at \space 0.995) \] \[ \mathcal{X}_{R}^{2} = 191.036 \space (value \space at \space 0.005) \]

03

Calculate Confidence Interval for \(\sigma^2\)

The confidence interval for \(\sigma^2\) can be calculated using the formula: \[ \left( \frac{(n-1)s^2}{\mathcal{X}_{R}^2}, \frac{(n-1)s^2}{\mathcal{X}_{L}^2} \right) \] Given: \( n = 153 \), \( s = 7.10 \mathrm{cm} \), and previously calculated \( df = 152 \) \[ \text{CI}(\sigma^2) = \left( \frac{152 \cdot (7.10)^2}{191.036}, \frac{152 \cdot (7.10)^2}{118.528} \right) \] \[ = \left( \frac{7630.84}{191.036}, \frac{7630.84}{118.528} \right) \] \[ = (39.93, 64.39) \]

04

Calculate Confidence Interval for \(\sigma\)

Take the square root of the confidence interval limits for \(\sigma^2\) to obtain the confidence interval for \(\sigma\). \[ \text{CI}(\sigma) = \left( \sqrt{39.93}, \sqrt{64.39} \right) \] \[ \approx (6.32, 8.02) \]

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

degrees of freedom

In statistics, the concept of 'degrees of freedom' is essential in various analyses. It signifies the number of values in a calculation that are free to vary. For example, if you have 153 observations (sample size), the sum of deviations from the mean must be zero. This imposes one constraint, thus reducing the degrees of freedom by one. Therefore, with 153 observations, your degrees of freedom (often denoted as df) is 152. The formula to compute it is: \[ df = n - 1 \] It is a fundamental concept when determining how much independent information you have in your data set. In the given problem, df is calculated as follows:

  • Given: n = 153
  • df = 153 - 1 = 152
chi-square distribution

The chi-square distribution is a key probability distribution in statistics, primarily used for hypothesis testing and constructing confidence intervals for variances and standard deviations. This distribution is defined by the degrees of freedom and varies with the sample size. A chi-square ( \(\chi^2\) ) distribution is not symmetric, and its shape depends on the degrees of freedom. In the provided exercise, two critical values from the chi-square distribution table are required: one for the 0.005 percentile and the other for the 0.995 percentile, specifically for 152 degrees of freedom. These values represent the tails where a very small amount of the data (0.5%) lies. Here are the values:

  • Left critical value ( \(\mathcal{X}_{L}^{2}\) ): 118.528
  • Right critical value ( \(\mathcal{X}_{R}^{2}\) ): 191.036

These values serve as critical points in our confidence interval estimation.

confidence interval

The confidence interval provides a range of values within which we can expect a population parameter to lie with a certain level of confidence. To estimate a confidence interval for the standard deviation \(\sigma\) , the problem utilizes the sample size, standard deviation, degrees of freedom, and chi-square critical values. The steps to compute the confidence interval for \(\sigma\) are:

  • Calculate the confidence interval for the variance \(\sigma^2\) using the formula: \(\left( \frac{(n-1)s^2}{\mathcal{X}_{R}^2}, \frac{(n-1)s^2}{\mathcal{X}_{L}^2} \right)\)
  • For \( n = 153, s = 7.10 \mathrm{cm}, \text{df} = 152 \)
  • \(\text{CI}(\sigma^2) = \left( \frac{152 \cdot (7.10)^2}{191.036}, \frac{152 \cdot (7.10)^2}{118.528} \right) \)
  • \( = \left( \frac{7630.84}{191.036}, \frac{7630.84}{118.528} \right) \)
  • \( = (39.93, 64.39) \)

Finally, take the square root of these bounds to get the confidence interval for \(\sigma\):

  • \( \text{CI}(\sigma) = \left( \sqrt{39.93}, \sqrt{64.39} \right) \)
  • \( \approx (6.32, 8.02) \)

Thus, with 99% confidence, the population standard deviation \(\sigma \) is estimated to lie between 6.32 cm and 8.02 cm.

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Problem 8 Finding Critical Values and Conf... [FREE SOLUTION] (3)

Most popular questions from this chapter

Use the data and confidence level to construct a confidence interval estimateof \(p,\) then address the given question. One of Mendel's famous genetics experiments yielded 580 peas, with 428 of themgreen and 152 yellow. a. Find a \(99 \%\) confidence interval estimate of the percentage of greenpeas. b. Based on his theory of genetics, Mendel expected that \(75 \%\) of theoffspring peas would be green. Given that the percentage of offspring greenpeas is not \(75 \%,\) do the results contradict Mendel's theory? Why or whynot?Finding Critical Values and Confidence Intervals. In Exercises \(5-8,\) use thegiven information to find the number of degrees of freedom, the criticalvalues \(\mathcal{X}_{L}^{2}\) and \(\mathcal{X}_{R}^{2},\) and the confidenceinterval estimate of \(\boldsymbol{\sigma} .\) The samples are from Appendix\(\boldsymbol{B}\) and it is reasonable to assume that a simple random samplehas been selected from a population with a normal distribution. Platelet Counts of Women \(99 \%\) confidence; \(n=147, s=65.4\)
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Problem 8 Finding Critical Values and Conf... [FREE SOLUTION] (2024)

FAQs

How to find critical value given confidence level and degrees of freedom? ›

Step 1: Express the confidence level as a number (decimal) with 0 < c < 1 . Step 2: Obtain the significance level, denoted , by α = 1 − c . Step 3: Use the -table or a calculator to obtain the -score (critical value) t α / 2 where (i) the is from Step 2 and (ii) the degrees of freedom equals , where is the sample size.

What is the critical value for a 99.5 confidence level? ›

Short Answer

The critical value z α 2 for 99.5% level of confidence is 2.81.

What is the critical value for a 98% level of confidence? ›

Student's T Critical Values
Conf. Level50%98%
One Tail0.2500.010
900.6772.368
1000.6772.364
z0.6742.326
36 more rows

What is the t-critical value for a 95% confidence interval? ›

Table of Critical t-Values for 95% Confidence Level
ν = n - 1tcrit
112.706
24.303
33.182
42.776
14 more rows

What is the formula for the degree of freedom critical value? ›

Express the critical probability of 97.5% as the t statistic like this:Degree of freedom (df) = the sample size - 1. This means that the number of samples you have in your study subtracted by one will equal the degree of freedom.

What is the critical value zα 2 that corresponds to a 93% confidence level? ›

Using a standard normal distribution table or calculator, we can find that the z-score that encloses 93% of the area under the curve is approximately 1.81. Therefore, the critical value zα/2 that corresponds to a 93% level of confidence is 1.81 (rounded to two decimal places). Answer: 1.81.

What is the critical value for 97% confidence? ›

Thus, the crucial value of z for a 97% confidence interval is 2.17, as determined by a z score table, which is as follows: Therefore the obtained probability for the z-score of 2.17 is 0.97. The answer is given above.

How to find the critical value corresponding to a 95% confidence level? ›

Step 1: Determine the confidence level, denoted , where C is a number (decimal) between 0 and 100. The confidence level is 95%. Step 2: Obtain the confidence level, denoted by evaluating α = 1 − C 100 . α = 1 − 95 100 = 1 − 0.95 = 0.05 .

What is the critical value for a confidence level of 92%? ›

Therefore, the critical value for a 92% confidence interval, rounded to three decimal places, is 1.751.

How to use t table to find critical value? ›

Finding the critical value t∗ given a significance level α , using the table with critical t values
  1. Find the row with the appropriate number of degrees of freedom (df)
  2. Find the column for the upper tail probability equal to α/2.
  3. At the intersection point of this row and column, you find the positive critical value t∗ .

How to find critical z value given confidence level? ›

Step 1: Determine the confidence level, denoted , where is a number (decimal) between 0 and 100. Step 2: Obtain the confidence level, denoted by evaluating α = 1 − C 100 . Step 3: Use the -table (or a calculator) to obtain the -score z α / 2 .

What is the critical value of F at 95% confidence? ›

The critical value for a 95% confidence interval is 1.96, where (1-0.95)/2 = 0.025.

What is the critical value for 95 confidence interval with DF? ›

For a 95% confidence interval Zcritical = 1.96. The problem states a sample size of 30, so we will use use t-critical with 30-1=29 degrees of freedom. A 95% confidence interval would encompass all but the bottom 2.5% and the top 97.5% which correspond to probabilities of 0.025 and 0.975.

What is the formula for the critical value of a confidence interval? ›

Multiply the critical value of t by s/√n. Add this value to the mean to calculate the upper limit of the confidence interval, and subtract this value from the mean to calculate the lower limit.

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