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| Section | Objectives |
|---|---|
| Probability Theory | - Fundamental Probability Concepts
|
| Statistical Inference | - Hypothesis Testing (Introductory Level)
|
| Regression and Modeling | - Linear Relationships
|
| Descriptive Statistics | - Single Variable Data Analysis
|
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80. Frage
Probability of drawing red card from deck = ?
Antwort: D
Begründung:
A standard deck has 52 cards divided into four suits: hearts, diamonds, clubs, and spades. Hearts and diamonds are red, while clubs and spades are black. Each suit has 13 cards, so the number of red cards is 13 +
13 = 26. The probability of drawing a red card is therefore the number of favorable outcomes divided by the total number of outcomes: 26/52. This simplifies to 1/2. Option B, 1/4, would describe the probability of drawing one specific suit, such as hearts only. Option C, 1/13, would describe one specific rank in the deck, such as a particular value across suits. Option D does not correspond to the card structure of a standard deck.
Since exactly half the deck is red, the correct probability is 26/52 = 1/2. Study Guide references/topics: card probability, favorable outcomes, sample space, theoretical probability.
81. Frage
Z-score = ?
Antwort: C
Begründung:
A z-score standardizes a raw value by expressing its distance from the mean in standard deviation units. The formula is z = (x # #)/#, where x is the observed value, # is the mean, and # is the standard deviation. The numerator x # # measures how far the value is from the mean. Dividing by # converts that distance into standard deviation units. A positive z-score means the value is above the mean, a negative z-score means the value is below the mean, and z = 0 means the value equals the mean. Option B divides the raw value by the mean and does not measure standardized distance. Option C reverses the ratio incorrectly. Option D gives only the raw deviation and fails to standardize by the standard deviation. Z-scores are central in normal distribution calculations and comparisons across different scales. Study Guide references/topics: z-score, standardization, mean, standard deviation.
82. Frage
A team of scientists wants to understand the relationship between smoking and lung cancer. They collect data from 1,000 individuals, recording their smoking habits and whether they have been diagnosed with lung cancer. The scientists then analyze the data to find any correlations.
Is this study observational or experimental?
Antwort: B
Begründung:
This is an observational study because the scientists collect existing information about individuals' smoking habits and lung cancer diagnoses without assigning treatments or controlling exposure. In an experimental study, researchers impose a condition, treatment, or intervention and then measure the resulting effect. Here, the researchers are not requiring some individuals to smoke and others not to smoke; they are simply recording naturally occurring characteristics and outcomes. The purpose is to examine association, or correlation, between two variables: smoking status and lung cancer diagnosis. Both variables are categorical, so the analysis would likely involve proportions, conditional percentages, or contingency tables. Option A is incorrect because analyzing correlations does not make a study experimental. Option B is incorrect because
"testing effects" would require manipulation or assignment. Option C identifies a true feature, a large sample size, but sample size does not determine study type. References/topics from the Study Guide: observational studies, experimental studies, association, categorical variables.
83. Frage
Review the following inequality:
y # #4x + 1
Which shaded region is described by this inequality?
Antwort: D
Begründung:
The inequality y # #4x + 1 has boundary line y = #4x + 1. The slope is #4, so the line must decrease steeply from left to right. The y-intercept is 1, so the line must cross the vertical axis at y = 1. Because the inequality uses "#," the boundary line must be solid, meaning points on the line are included in the solution set. The solution region must be below the line because y is less than or equal to the expression #4x + 1. A quick test point confirms this. Substitute (0, 0): 0 # #4(0) + 1, giving 0 # 1, which is true. Therefore, the shaded solution region must include the origin. Graph D shows a solid, steep downward-sloping line crossing the y-axis near 1 and shades the region containing the origin and below the boundary. References/topics from the Study Guide:
linear inequalities, slope-intercept form, boundary lines, test-point method.
84. Frage
There are 10 green, 20 yellow, and 15 purple marbles in a jar.
What is the probability of selecting 2 green marbles in a row and then 1 yellow marble, replacing the marble each time?
Antwort: D
Begründung:
There are 10 green, 20 yellow, and 15 purple marbles, so the total number of marbles is 10 + 20 + 15 = 45.
Because each marble is replaced after selection, the probabilities remain the same on every draw, making the events independent. The probability of selecting a green marble on one draw is 10/45 = 2/9. Selecting two green marbles in a row gives (2/9)(2/9). The probability of then selecting a yellow marble is 20/45 = 4/9.
Multiply the probabilities because the question asks for green and green and yellow in that exact sequence: (2
/9)(2/9)(4/9) = 16/729. Replacement is essential; without replacement, the denominators and numerators would change after each draw. Since replacement keeps the sample space constant at 45 marbles each time, the independent multiplication rule applies directly. References/topics from the Study Guide: independent events, replacement, multiplication rule, compound probability.
85. Frage
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