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The mean SAT score of all students who took the test recently was 1020.
The standard deviation of SAT scores for students taking the test with Debbie is 153.
Debbie can determine her required score by finding the z-score that corresponds to the top 10% of the normal distribution, which is approximately 1.28. She can then use the formula x = μ + zσ, where μ is the mean and σ is the standard deviation.
Debbie should aim for a score of approximately 1216 on the SAT to ensure that only about 10% of examinees score higher than she does.
The z-score indicates how many standard deviations an element is from the mean. In this context, it helps determine the score that corresponds to a specific percentile in the distribution of SAT scores.
The area to the left of the x value for the top 10% of scores is 0.9000, which represents the cumulative probability of scoring below that value.
The normal distribution table is used to find the z-score that corresponds to a specific cumulative probability, which helps in calculating the required score for a desired percentile.
The formula used to calculate Debbie's required SAT score is x = μ + zσ, where μ is the mean score, z is the z-score, and σ is the standard deviation.
The probability that a randomly selected college student has credit card debt between $2109 and $3605 is approximately 61.36%.
The mean credit card debt for college students is $3173, and the standard deviation is $800.
To calculate the z-scores, use the formula z = (x - μ) / σ. For $2109, z = (2109 - 3173) / 800 = -1.33. For $3605, z = (3605 - 3173) / 800 = 0.54.
The cumulative probability for a z-score of -1.33 is approximately 0.0918.
The cumulative probability for a z-score of 0.54 is approximately 0.7054.
To find the probability of a range of values in a normal distribution, calculate the z-scores for the endpoints of the range, find the cumulative probabilities for those z-scores, and then subtract the lower cumulative probability from the upper cumulative probability.
The normal approximation of the binomial distribution is important because it allows for easier calculations and analysis of probabilities for large sample sizes, where the binomial distribution can be complex.
The four conditions for a binomial distribution are: 1) Each trial results in one of two outcomes (success or failure), 2) The probability of success is constant for each trial, 3) The trials are independent, and 4) There is a fixed number of trials.
Independence in binomial trials means that the outcome of one trial does not affect the outcome of another trial, ensuring that the probability of success remains constant across trials.
The mean indicates the center of the distribution, while the standard deviation measures the spread or variability of the data around the mean, helping to understand the distribution's shape and probabilities.
Knowing her target SAT score is important for Debbie to set a realistic goal for her preparation, understand the competitive landscape, and increase her chances of being accepted into her desired college.
Statistical methods such as descriptive statistics, z-scores, normal distribution analysis, and probability calculations can be used to analyze SAT scores and credit card debt.
Understanding normal distributions can help students interpret test scores, assess their performance relative to peers, and make informed decisions about their study strategies and academic goals.