A delivery service tracks the weights of its packages. A sample of 20 packages has a variance of 4.5 lbs 2 . Construct a 95% conf. int. for the population variance. Assume a normal distribution.

So now let's get a little bit more practice building a confidence interval for variance. In this next practice problem, we have a delivery service that tracks the weights of its packages. Now a sample of 20 packages has a variance sample variance of 4.5 pounds squared. We're gonna construct a 95% confidence interval for the population variance, and we're gonna assume that the sample comes from a population that is normally distributed. This is sort of just a disclaimer that tells us that we're allowed to do this step. Alright? Now as a reminder, we have this formula that basically tells us directly and builds the confidence interval for us. We have our s squared, the sample variance, that's our best guess. But instead of calculating in a margin of error and then doing a s squared plus or minus e, we actually plug in the s squared directly into these formulas with our chi squared values to find this confidence interval. Alright? So what we're going to see here is the first part of this process is actually very similar to what we've previously done before when we were just finding chi squared values. We're going to need an alpha. We're going to need an alpha over two, one minus alpha over two. We'll need a degrees of freedom, and then that will basically tell us we'll use our chi squared tables to find a chi squared r and l. Alright? So let's actually go ahead and do all of those things, And then later on, we'll see how we can just sort of build all of this in our confidence interval. Alright? So we have our confidence level of 95%. That's gonna be our c, which means that our alpha is gonna be one minus that number expressed as a decimal. That's 0.05. That means that if we take that area and cut it in half, we're gonna get an alpha over two of 0.025. We'll use that number to find chi squared r, and then we'll use one minus alpha over two, zero point nine seven five in order to figure out our chi squared l. Alright. In order to do that, we'll need our degrees of freedom, which is just going to be, as a reminder, the sample size minus one. We have a sample of 20 packages, meaning our degrees of freedom will be 19. Alright. Now we're just going to go ahead and move over to our chi squared table and find those values. Alright? So we're going to go ahead and use these two columns over here. We have zero point zero two five and nine seven five. We just go all the way down until we hit a degrees of freedom of 19. We should just be using these two numbers right over here, eight point nine zero seven and thirty two point eight five two. I've go ahead and just rounded those, to the nearest hundreds place in order to use them. Alright? So let's go back to our problem over here. We've got a chi squared r, 32.85, chi squared l, 8.91. Alright? Now, again, we use these two numbers, and we basically just plug them directly into this formula over here, and that just gives us what our confidence interval is. If you remember that the chi squared on the right is gonna be the one that you use for the lower bounds, and the l is gonna be for the upper bound. They basically get switched, and it's because this chi squared value ends up being higher if it's in the denominator and ends up being a lower number. That's the reason why. You don't really have to memorize that. If you just go ahead and plug them in, you'll see that it worked out this way. Alright? So we're gonna have n minus one, which again is just the degrees of freedom. So this is another words where it's gonna be 19 times s squared. Our s squared just comes directly from the problem, and it's 4.5. Really important. This number is already 4.5. You don't have to square it again. So when you plug it into this formula, you do not square that number. That's a really common mistake that most people make. The sample variance, s squared, is already 4.5, so you'd leave just leave that number alone. Alright? So we're gonna divide now by chi squared r, which is gonna be 32.85. And then this is gonna be less than our s our sigma squared, and this is gonna be again, the numerator stays the same, 19 times 4.5, but now we divide by our chi squared l, which is gonna be our 8.91. Okay? So if you go ahead and work out both of these numbers, this should just build out your confidence interval, and those two numbers end up being 2.6 on the left. So that's less than sigma squared, and that's less than 9.6 on the right side. Alright? So we have that the sigma were 95% confident that the true population variance sigma squared lies somewhere between these two numbers, two point sixty and nine point sixty. Now if you wanna just leave this expressed in this, sort of notation like this, that is perfectly fine. This is a perfectly perfectly acceptable way to express your confidence intervals. However, in some courses where you might use bracket notation, you could have also written this as, 2.6 comma 9.6 as well. So both of those are perfectly reasonable. Just depends on what you're required to do for your course. But that's really all there is to it. So let me know if you have any questions, and let's move on.

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1. Introduction to Statistics53m

Intro to Stats
22m

Intro to Collecting Data
8m

Sampling Methods
23m

2. Describing Data with Tables and Graphs2h 2m

Visualizing Qualitative vs. Quantitative Data
4m

Frequency Distributions
35m

Histograms
7m

Histograms w/ Graphing CalculatorBonus
13m

Bar Graphs and Pareto Charts
11m

Pie Charts
9m

Frequency Polygons
10m

Dot Plots
6m

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14m

Time-Series Graph
9m

3. Describing Data Numerically2h 8m

Mean
9m

Median
20m

Mode
7m

Standard Deviation
16m

Interpreting Standard Deviation
20m

Percentiles & Quartiles
14m

5-Number Summary Using a TI-84Bonus
10m

Boxplots
8m

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11m

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8m

4. Probability2h 26m

Basic Concepts of Probability
7m

Complements
6m

Addition Rule
17m

Multiplication Rule: Independent Events
11m

Introduction to Contingency Tables
17m

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21m

Bayes' Theorem
13m

Fundamental Counting Principle
8m

Counting
41m

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Discrete Random Variables
38m

Binomial Distribution
1h 17m

Finding Binomial Probabilities-ExcelBonus
17m

Poisson Distribution
45m

Finding Poisson Probabilities-ExcelBonus
15m

Hypergeometric Distribution
14m

6. Normal Distribution & Continuous Random Variables2h 21m

Uniform Distribution
19m

Standard Normal Distribution
39m

Probabilities & Z-Scores w/ Graphing CalculatorBonus
19m

Non-Standard Normal Distribution
30m

Finding Probabilities, Z Values, and X Values with the Normal Distribution-ExcelBonus
32m

7. Sampling Distributions & Confidence Intervals: Mean3h 37m

Sampling Distribution of the Sample Mean and Central Limit Theorem
19m

Distribution of Sample Mean - ExcelBonus
23m

Introduction to Confidence Intervals
22m

Confidence Intervals for Population Mean
1h 26m

Determining the Minimum Sample Size Required
12m

Finding Probabilities and T Critical Values - ExcelBonus
28m

Confidence Intervals for Population Means - ExcelBonus
25m

8. Sampling Distributions & Confidence Intervals: Proportion2h 20m

Sampling Distribution of Sample Proportion
35m

Confidence Intervals for Population Proportion
45m

Confidence Intervals for Population Proportion - ExcelBonus
12m

Chi Square Distribution
20m

Confidence Intervals for Population Variance
26m

9. Hypothesis Testing for One Sample5h 15m

Steps in Hypothesis Testing
1h 13m

Performing Hypothesis Tests: Means
1h 1m

Hypothesis Testing: Means - ExcelBonus
42m

Performing Hypothesis Tests: Proportions
39m

Hypothesis Testing: Proportions - ExcelBonus
27m

Performing Hypothesis Tests: Variance
12m

Critical Values and Rejection Regions
29m

Link Between Confidence Intervals and Hypothesis Testing
12m

Type I & Type II Errors
16m

10. Hypothesis Testing for Two Samples5h 35m

Two Proportions
1h 12m

Two Proportions Hypothesis Test - ExcelBonus
28m

Two Means - Unknown, Unequal Variance
1h 2m

Two Means - Unknown Variances Hypothesis Test - ExcelBonus
12m

Two Means - Unknown, Equal Variance
15m

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9m

Two Means - Known Variance
12m

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21m

Two Means - Matched Pairs (Dependent Samples)
42m

Matched Pairs Hypothesis Test - ExcelBonus
12m

Two Variances and F Distribution
29m

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15m

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26m

Correlation Coefficient
21m

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6m

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17m

Inferences for the Correlation Coefficient - ExcelBonus
11m

12. Regression3h 42m

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26m

Residuals
12m

Coefficient of Determination
12m

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8m

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11m

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32m

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1m

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21m

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13m

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19m

Multiple Regression - ExcelBonus
29m

Quadratic Regression
23m

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10m

13. Chi-Square Tests & Goodness of Fit2h 31m

Goodness of Fit Test
50m

Goodness of FIt Test Using TI-84Bonus
18m

Goodness of Fit Test - ExcelBonus
10m

Contingency Tables
13m

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14m

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11m

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6m

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12m

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13m

14. ANOVA2h 32m

Introduction to ANOVA
34m

One-Way ANOVA - ExcelBonus
12m

Multiple Comparisons: Tukey Test
14m

Multiple Comparisons: Tukey-Kramer Test
15m

Multiple Comparisons: Bonferoni Test
24m

Two-Way ANOVA
32m

Two-Way ANOVA - ExcelBonus
18m

8. Sampling Distributions & Confidence Intervals: Proportion

Confidence Intervals for Population Variance

Statistics for Business

8. Sampling Distributions & Confidence Intervals: Proportion

Confidence Intervals for Population Variance

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Multiple Choice

A delivery service tracks the weights of its packages. A sample of 20 packages has a variance of 4.5 lbs². Construct a 95% conf. int. for the population variance. Assume a normal distribution.

(2.60, 9.60)

(2.74, 9.60)

(2.99, 8.89)

(2.84, 8.45)

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Verified step by step guidance

Identify the sample size (n = 20) and the sample variance (s^2 = 4.5). Since the population is assumed to be normally distributed, we can use the Chi-square distribution to construct the confidence interval for the population variance.

Determine the degrees of freedom, which is given by \( df = n - 1 = 20 - 1 = 19 \).

Find the critical Chi-square values for a 95% confidence interval. These are \( \chi^2_{\alpha/2, df} \) and \( \chi^2_{1-\alpha/2, df} \), where \( \alpha = 0.05 \). You can find these values from a Chi-square distribution table or statistical software.

Use the formula for the confidence interval for the population variance \( \sigma^2 \): \[ \left( \frac{(n-1)s^2}{\chi^2_{1-\alpha/2, df}}, \quad \frac{(n-1)s^2}{\chi^2_{\alpha/2, df}} \right) \] Substitute the values of \( n-1 \), \( s^2 \), and the critical Chi-square values into this formula.

Calculate the lower and upper bounds of the confidence interval using the formula above. This interval estimates the range in which the true population variance lies with 95% confidence.