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What is the significance of the z-score in determining an exam score percentile?
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The z-score represents the relative position of a particular score within the distribution, allowing you to find the corresponding percentile.
How is skewness indicated in a distribution, and what does a left-skewed distribution imply about its mean and median?
A left-skewed distribution means that the mean is less than the median, indicating a disproportional number of low values.
How do you calculate the interquartile range (IQR) of a data set?
Subtract the first quartile (Q1) from the third quartile (Q3). For example, if Q3 is 12.85 and Q1 is 8.96, then IQR = 12.85 - 8.96 = 3.89.
How do you calculate the proportion of days selling less than 175.32 gallons of milk if the mean is 194.35 gallons and the standard deviation is 22.74 gallons?
First, calculate the z-score: (175.32 - 194.35) / 22.74 = -0.84. Then look up -0.84 in the z-table to find the proportion, which is approximately 0.2005.
Describe the empirical rule and how it applies to calculating intervals for normal distributions.
The empirical rule states that about 68% of data within a normal distribution falls within one standard deviation of the mean. For a mean of 125.25 and SD of 13.59, the interval is [111.66, 138.84].
For a multiple-choice test with 50 questions and a probability of 0.25 for a correct answer, how do you find the probability of answering between 9 and 11 questions correctly?
Use the continuity correction and normal approximation for the binomial distribution, calculating the z-scores for 9 and 11 correctly answered questions.
What is the z-score formula for a normal distribution, and how do you use it?
The formula is z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation. It is used to find the position of a value within a normal distribution.
How is standard error calculated and how can it be reduced?
Standard error is calculated by sqrt(p(1-p)/n). To decrease it by a factor of 5, you would need to increase the sample size by a factor of 25.
Under what circumstances can a histogram indicate left skewness?
A histogram indicates left skewness when it has a longer tail on the left side, suggesting that a majority of data points are concentrated on the higher end.
What conditions are required for the Central Limit Theorem to apply to a sample mean distribution?
The sample size should be sufficiently large (usually n ≥ 30) for the distribution of the sample mean to approximate a normal distribution, regardless of the shape of the population distribution.
In what ways can you determine the probability of a business audit rate being less than expected?
Calculate the z-score for the observed audit rate, compare it against the standard normal distribution, and interpret the resulting probability.
How would you determine the new mean fuel economy if the current mean is 23.86 mpg with an increase of 5.49 mpg per observation?
Add the increase to the current mean: 23.86 + 5.49 = 29.35 mpg. The standard deviation remains unchanged.
How do categorical variables differ from other types of variables, and can you give examples?
Categorical variables classify data into distinct categories without a numerical order. Examples include employee category and income tax bracket.
What formula is used to calculate the standard error of a population proportion, and how would you apply it to a survey?
The standard error formula for a population proportion is sqrt(p(1-p)/n), where p is the sample proportion and n is the sample size. For 122 out of 148 respondents: sqrt(0.824(1-0.824)/148) = 0.033.
In probability terms, what does a z-score of 3.36 mean for rolling a 3 more than 50 times out of 195 die rolls?
A z-score of 3.36 indicates a very low probability of occurrence, as it corresponds to a probability of 0.0004.
What does the probability of selling fewer drinks than the average indicate, and how is it calculated?
It indicates the likelihood of a specific outcome being below the average. Calculate the z-score using the mean and standard deviation, then find the probability using a z-table.
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