Final answer:
To calculate probabilities of credit card debt amounts for college seniors, the Z-score formula is used, with the mean debt at $3262 and standard deviation of $1100. The probabilities for owing at least $1000 (already provided), more than $4000, and between $3000 and $4000 can be found using the standard normal distribution or a calculator.
Step-by-step explanation:
Probability Calculations for Credit Card Debt
The student's question involves calculating probabilities for college seniors' credit card debt, assuming a normal distribution. With an average (mean) debt of $3262 and a standard deviation of $1100, we can calculate the following probabilities:
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- a) The probability that a senior owes at least $1000.
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- b) The probability that a senior owes more than $4000.
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- c) The probability that a senior owes between $3000 and $4000.
To find these probabilities, we use the Z-score formula: Z = (X - μ) / σ, where X is the value for which we're finding the probability, μ is the mean, and σ is the standard deviation. We then refer to the standard normal distribution table or use a calculator to find the corresponding probabilities.
Calculations
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- For a senior owing at least $1000, we find the Z-score and the area to the right of it, which is approximately 0.9803.
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- For a senior owing more than $4000, we calculate the Z-score for $4000 and find the tail probability to the right.
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- For a senior owing between $3000 and $4000, we find the Z-scores for both values and calculate the area between them.
It is noted that the result for part a) was already given as 0.9803. For parts b) and c), similar calculations will yield the desired probabilities.