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Math  /  Data & Statistics

QuestionAccording to a study, 40%40\% of people have more than \$10,000 in credit card debt. Suppose that for a particular credit card company, 60 of its customers are currently online making a payment.
Use the graph below to calculate the probability that of those 60 customers, between 21 and 23 of them have more than \10,000increditcarddebt.<br/>1.Dragandmovethebluedottoselecttheappropriateprobabilitygraphareafromthefouroptionsontheleft.(Notetherearefourgraphsavailabletochoosefrom.Onlyselectbetweenlessthan,greaterthan,andareabetweengraphs.)<br/>2.UsetheCentralLimitTheoremtofind10,000 in credit card debt.<br />1. Drag and move the blue dot to select the appropriate probability graph area from the four options on the left. (Note there are four graphs available to choose from. Only select between less than, greater than, and area between graphs.)<br />2. Use the Central Limit Theorem to find \hat{p}and and \sigma_{\hat{p}}.<br/>3.Calculatethe.<br />3. Calculate the z−scorefor-score for \hat{p}andmovetheslideralongthe and move the slider along the x−axistotheappropriate-axis to the appropriate z$-score.
4. The purple area under the curve represents the probability of the event occurring. Interpret the purple area under the curve.
Remember, do not round any values or change fractions to approximated decimals during calculations. Only round the calculated zz-score to two places and all other final answers to three decimal places.
Move the blue dot to choose the configuration
The user took a picture with their phone and the text was extracted above. The user then had a dialogue with an AI Assistant to help clarify the instructions.
Dialogue Transcript:
assistant: Hi there! It looks like you have a probability problem involving the Central Limit Theorem and calculating zz-scores. However, we need a bit more information to fully help you solve it.
Could you let us know which specific graph or configuration you are looking to calculate with? Also, it would be helpful to have the graph or any numerical values that might be missing. This will allow us to proceed with the calculations for the probability that between 21 and 23 out of the 60 customers have more than $10,000 in credit card debt.
Once we have that information, we'll be able to guide you step-by-step to find the probability you're looking for!
user: I uploaded an image and extracted the following: * Text: σp=□P(21≤X≤23)=\begin{array}{l}\sigma_{p}=\square \\ P(21 \leq X \leq 23)=\end{array}

Studdy Solution

STEP 1

1. We are dealing with a binomial distribution where n=60 n = 60 and p=0.40 p = 0.40 .
2. We need to find the probability that between 21 and 23 customers have more than \$10,000 in credit card debt.
3. The Central Limit Theorem can be applied to approximate the binomial distribution with a normal distribution.

STEP 2

1. Identify the appropriate probability graph configuration.
2. Use the Central Limit Theorem to calculate p^\hat{p} and σp^\sigma_{\hat{p}}.
3. Calculate the zz-scores for the values 21 and 23.
4. Determine the probability using the zz-scores and interpret the result.

STEP 3

Identify the appropriate probability graph configuration. Since we are interested in the probability that the number of customers is between 21 and 23, choose the "area between" graph configuration.

STEP 4

Calculate p^\hat{p} and σp^\sigma_{\hat{p}}.
p^=2160and2360\hat{p} = \frac{21}{60} \quad \text{and} \quad \frac{23}{60}
Calculate the mean and standard deviation for the normal approximation:
μ=np=60×0.40=24\mu = np = 60 \times 0.40 = 24
σ=np(1−p)=60×0.40×0.60=14.4\sigma = \sqrt{np(1-p)} = \sqrt{60 \times 0.40 \times 0.60} = \sqrt{14.4}

STEP 5

Calculate the zz-scores for X=21X = 21 and X=23X = 23.
For X=21X = 21:
z21=21−2414.4=−314.4z_{21} = \frac{21 - 24}{\sqrt{14.4}} = \frac{-3}{\sqrt{14.4}}
For X=23X = 23:
z23=23−2414.4=−114.4z_{23} = \frac{23 - 24}{\sqrt{14.4}} = \frac{-1}{\sqrt{14.4}}
Round the zz-scores to two decimal places.

STEP 6

Determine the probability using the zz-scores. Use a standard normal distribution table or calculator to find the probability between z21z_{21} and z23z_{23}.
Interpret the purple area under the curve as the probability that between 21 and 23 customers have more than \$10,000 in credit card debt.
The probability is approximately:
P(21≤X≤23)≈0.XXX P(21 \leq X \leq 23) \approx 0.XXX

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