Probability and Normal Distribution

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Questions and Answers

Match the components of the z-score formula with their descriptions:

x = Value from the dataset μ = Mean of the distribution σ = Standard deviation of the distribution z = Z-score representing the standard score

Match the terms used in normal distribution with their definitions:

Mean (μ) = Average of all data points Standard deviation (σ) = Measure of dispersion in the dataset Top 15% = Percentage of highest marks Z-score = Standardized score indicating how many standard deviations away a value is

Match the steps in determining the lowest mark for an A with their purposes:

Step 1 = Define the variable and what x represents Step 2 = Identify the mean and standard deviation Step 4 = Graph the normal distribution Step 6 = Find the corresponding z-score using probability

Match the elements involved in calculating the lowest mark for an A:

Final marks = Normally distributed marks among students Top 15% = Marks that categorize students as top achievers Mean of 72 = Average mark for the cohort Standard deviation of 5 = Variation in student marks Signup and view all the answers

Match the values with their corresponding variable in the context of Mr. Tan's study:

x = Lowest mark needed for an A μ = 72 marks (mean score) σ = 5 marks (standard deviation) P(x > x1) = Probability of achieving above the lowest mark for an A Signup and view all the answers

Match the following values to their implications in different standard deviations:

$x = 76$ in (a) = One of the highest results in the cohort $x = 76$ in (b) = Slightly above average in the cohort Standard deviation $\sigma = 3$ in (a) = Close clustering around the mean Standard deviation $\sigma = 12$ in (b) = Wider spread of data points Signup and view all the answers

Match the empirical rule percentages to the respective ranges in standard deviations:

68% Rule = Within one standard deviation of the mean 95% Rule = Within two standard deviations of the mean 99.7% Rule = Within three standard deviations of the mean 0% Rule = Outside three standard deviations of the mean Signup and view all the answers

Match the following definitions to their corresponding empirical rule:

68% Rule = Most data points cluster closely to the mean 95% Rule = Most values fall further out from average 99.7% Rule = Nearly all data points covered Standard deviation = Measure of data spread around the mean Signup and view all the answers

Match the following concepts related to raw data with their descriptions:

Raw data = Difficult to interpret and compare Relative location = Depends on mean and standard deviation Single value = Does not provide full context within distribution Interpretation of data = Requires understanding of distribution characteristics Signup and view all the answers

Match the following scenarios with their corresponding rules in the normal distribution:

68% of heights = Close to average height, neither too tall nor too short 95% of heights = Most people falling slightly away from average 99.7% of heights = Includes all but extreme cases 100% of heights = Encompasses everyone in the dataset Signup and view all the answers

Match the following statistical terms to their definitions:

Standard deviation ($\sigma$) = Measure of spread in data Cohort = Group being analyzed Mean ($ ext{µ}$) = Average value in the dataset Normal distribution = Bell-shaped curve representing data spread Signup and view all the answers

Match the following facts about distributions with their explanations:

Higher standard deviation = Wider spread of data points Lower standard deviation = Data points close to the mean Mean shifts = Affects data interpretation Data interpretation = Requires context from distribution characteristics Signup and view all the answers

Match the following terms to their relevance in everyday life:

Normal distribution = Applicable across various fields Empirical rule = Used for summarizing data spread Standard deviation = Helps understand variability Cohort analysis = Analyzes specific groups over time Signup and view all the answers

Match the following terms related to the normal distribution with their descriptions:

Mean = The average value in a data set Standard Deviation = Measure of dispersion from the mean Z-score = Standardized value indicating how many standard deviations an element is from the mean Symmetrical = Property meaning one half is a mirror image of the other Signup and view all the answers

Match the following symbols with their meanings in the context of statistics:

µ = Mean of a population σ = Standard deviation of a population p = Probability value x = Value in the data set Signup and view all the answers

Match the following probability statements with their corresponding z-scores:

p(x > 105) = p(z > 2.50) p(z < 0) = 0.5 p(z = 2.50) = 0.4938 p(z < -1) = Value less than 0.5 Signup and view all the answers

Match the following components of normal distribution steps with their appropriate actions:

Step 1 = Define the variable x Step 4 = Draw a normal distribution curve Step 6 = Convert x-value to z-score Step 8 = Refer to the Standard Normal Distribution table Signup and view all the answers

Match the following terms with their significance in the distribution of tips:

$80 = Mean amount of tips $10 = Standard deviation of tips $105 = Threshold amount of tips being analyzed 2.50 = Z-score for the tips over $105 Signup and view all the answers

Match the following components with their roles in a normal distribution:

Area under the curve = Total probability equals 1 Left half of the curve = Area to the left of the mean Right half of the curve = Area to the right of the mean Standard normal distribution = A normal distribution with mean 0 and standard deviation 1 Signup and view all the answers

Match the following statistical concepts with their characteristics:

Normal distribution = Symmetrical and bell-shaped Z-score transformation = Uses mean and standard deviation Probability table = Used to find area under the curve Dispersion = How spread out the values are Signup and view all the answers

Match the following terms related to z-scores with their correct definitions:

Mean = Average value from which z-scores are calculated Z-score = Standardized value indicating deviation from the mean Standard Normal Distribution = Distribution with mean of 0 and standard deviation of 1 Negative z-scores = Represents data below the mean Signup and view all the answers

Match the following statistical elements with their relevance to Elena's situation:

Housekeeping team tips = Normally distributed Mean ($80) = Expected amount of tips received Standard deviation ($10) = Variation in tips received Probability (p(z > 2.50)) = Chance of exceeding $105 in tips Signup and view all the answers

Match the following components of the z-score formula with their descriptions:

x = The raw score or data point being transformed μ = The mean of the dataset σ = The standard deviation of the dataset z = The computed z-score value Signup and view all the answers

Match the following steps with the correct order of finding probabilities using the Standard Normal Distribution Table:

Round the z-score = Compute z-score to two decimal places Locate the value = Find the left margin for the first part of z-score Find the column = Look at the top margin for the second part of z-score Read the probability = Identify the intersection of the row and column Signup and view all the answers

Match the following probabilities with their z-score conditions:

Probability = 0.50 = Area under the curve to the right of the mean Probability = 0.1985 = Area corresponding to a z-score of 0.52 Negative z-score = Area below the mean Positive z-score = Area above the mean Signup and view all the answers

Match the following parts of z-score calculation with their numerical examples:

x-value of 100 = Used when z = 0 μ = 100 = Mean in the context of z-scores σ = 10 = Standard deviation in the example provided z = 0 = Indicates data point is at the mean Signup and view all the answers

Match the following empirical rule statements with their corresponding probabilities:

Total area under the curve = Probability = 1.00 Area to the left of the mean = Probability = 0.50 Area to the right of the mean = Probability = 0.50 Height of normal curve = Symmetric around the mean Signup and view all the answers

Match the following aspects of z-scores with their characteristics:

Z-scores of 0 = Indicates the score is exactly at the mean Positive z-scores = Indicates scores above the mean Negative z-scores = Indicates scores below the mean Standard deviation of z-scores = Always equal to 1 Signup and view all the answers

Match the following z-score computations with their implications:

z = 0 = Data point is exactly at the mean z = 0.52 = Data point is above the mean, with a specific probability Negative z-score = Data point lies below the mean Positive z-score = Data point lies above the mean Signup and view all the answers

Match the following parameters of a normal distribution with their effects on the distribution curve:

Mean ($ ext{µ}$) = Determines the center of the distribution Standard Deviation ($ ext{σ}$) = Indicates the spread of the distribution Larger Standard Deviation = Wider, flatter curve Smaller Standard Deviation = Narrower, more peaked curve Signup and view all the answers

Match the following terms with their definitions related to distribution in samples:

Normal Distribution = A probability distribution that is symmetric around the mean Mean ($ ext{µ}$) = Average value of a dataset Standard Deviation ($ ext{σ}$) = A measure of the amount of variation or dispersion of a set of values Variable = A characteristic or attribute that can assume different values Signup and view all the answers

Match the following probabilities with their corresponding interpretations:

0.0062 = Probability that Elena receives more than $105 in tips 0.0668 = Probability of perceived poor service by hotel guests 0.8351 = Probability that Elena receives between $70 and $105 in tips 0.4938 = Probability of z-score being 2.50 Signup and view all the answers

Match the possible scores with their implications in comparison to the cohort mean:

Score of 76 = B+ grade indicating above average performance Cohort Mean of 70 = Below which most scores tend to cluster Cohort Mean of 85 = Above which indicates lower performance for the score of 76 Standard Deviation of 1.6 = Indicates consistency among competitors' scores Signup and view all the answers

Match the following z-scores with their respective probabilities:

z < -1.50 = 0.0668 z > 2.50 = 0.0062 -1.00 to 2.50 = 0.8351 z = 10 = 0.4938 Signup and view all the answers

Match the following statements about Koko Waffles' cereal distribution with their outcomes:

Different Mean Weights = Indicates varying product formulations Same Standard Deviation of 1.6 grams = Suggests consistency in weight across product lines Wider Curve = Indicates greater variability among data points Narrower Curve = Signifies scores are closely clustered around the mean Signup and view all the answers

Match the following x-values to their related calculations:

$105 = p(z > 2.50) $65 = p(z < -1.50) $70 = p(-1.00 ≤ z ≤ 2.50) $80 = Mean for probability calculations Signup and view all the answers

Match the scenarios with their corresponding effects on employee service years distribution:

Same Means in Offices = Indicates similar average years of service Different Standard Deviations = Results in varied shapes of the distribution curve Smaller Standard Deviation = Indicates data points are closely clustered around the mean Larger Standard Deviation = Results in greater variability among years of service Signup and view all the answers

Match the following steps with their statistical actions:

Finding p(x > $105) = $105 - 80 Finding p(x < $65) = $65 - 80 Calculating the probability = Using z-scores Interpreting results = Understanding the area under the normal distribution Signup and view all the answers

Match the following types of information with what helps evaluate scores in a cohort:

Absolute Score = Does not indicate relative performance Cohort Mean = Provides context for comparing individual performance Standard Deviation = Indicates how scores differ from the mean Comparison Across Cohorts = Reveals the significance of the score in a broader context Signup and view all the answers

Match the following consequences of a smaller standard deviation with their implications:

Narrow Distribution Curve = Indicates scores are tightly clustered around the mean Lower Variability = Suggests more consistency among scores Higher Predictability = Facilitates forecasting in business statistics Enhanced Reliability = Increases confidence in the data analysis Signup and view all the answers

Match the following phrases with their statistical context:

Probability under the curve = Related to normal distribution Shaded region probability = Represents area for z-scores Total tips less than $65 = Indicates poor service perception Total tips between $70 and $105 = Represents a confident service level Signup and view all the answers

Match the following cohorts with their likely characteristics based on mean performance:

Cohort with Mean of 70 = Indicates average performance Cohort with Mean of 85 = Indicates above-average competency Cohort with Below Average Scores = Indicates potential challenges in learning High-Performing Cohort = Suggests effective teaching methods Signup and view all the answers

Match the following conclusions with their corresponding scenarios:

0.0062 = Elena receiving more than $105 in tips 0.0668 = Perception of poor service at tips below $65 0.8351 = Receiving tips between $70 and $105 0.4938 = Probability related to z-score of 2.50 Signup and view all the answers

Match the following calculations with their result descriptions:

$105 - $80 = Calculating p(x > $105) $65 - $80 = Calculating p(x < $65) p(-1.00 ≤ z ≤ 2.50) = Finding probability between given values 0.50 - 0.4332 = Finding the probability for poor service perception Signup and view all the answers

Study Notes