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| Section | Objectives |
|---|---|
| Probability Distributions | - Continuous distributions
|
| Descriptive Statistics | - Data summarization
|
| Statistical Inference | - Estimation
|
| Probability | - Fundamental probability concepts
|
| Regression and Correlation | - Relationship analysis
|
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NEW QUESTION # 125
Determine z in the equation:
1/(6z) = 2/9
Answer: B
Explanation:
The equation 1/(6z) = 2/9 is a proportion because it sets two fractions equal to each other. To solve it, use cross multiplication. Multiply the numerator of the first fraction by the denominator of the second fraction, and multiply the denominator of the first fraction by the numerator of the second fraction: 1 × 9 = 2 × 6z. This gives 9 = 12z. To isolate z, divide both sides by 12: z = 9/12. This fraction can also be simplified to 3/4, but the answer choices list the unsimplified equivalent form 9/12. The other options do not satisfy the original equation. For example, if z = 2/54, then 6z is very small and the left side becomes much larger than 2/9. If z =
12/9 or 18/2, the denominator becomes too large, making the left side too small. References/topics from the Study Guide: proportions, solving equations, cross multiplication, equivalent fractions.
NEW QUESTION # 126
68-95-99.7 rule: % within 1 SD?
Answer: B
Explanation:
The 68-95-99.7 rule, also called the empirical rule, applies to approximately normal distributions. It states that about 68% of observations fall within 1 standard deviation of the mean, about 95% fall within 2 standard deviations, and about 99.7% fall within 3 standard deviations. The question asks specifically for the percentage within 1 standard deviation, so the correct answer is 68%. In notation, this interval is # # # to # +
#, where # is the mean and # is the standard deviation. This rule is useful for interpreting spread in bell- shaped distributions. For example, if scores have mean 80 and standard deviation 5, then approximately 68% of scores fall between 75 and 85. Option B corresponds to 2 standard deviations, and option C corresponds approximately to 3 standard deviations. Option D is not part of the empirical rule. Study Guide references
/topics: empirical rule, normal distribution, standard deviation, spread.
NEW QUESTION # 127
Central limit theorem applies when:
Answer: B
Explanation:
The central limit theorem applies when the sample size is sufficiently large and observations are independent.
It states that the sampling distribution of the sample mean becomes approximately normal as sample size increases, even if the original population distribution is not normal. This makes option A correct. Option B is incorrect because the population does not have to be normal for the theorem to operate; large sample size is what permits the normal approximation. Option C is not the condition that defines the theorem. A population parameter may be unknown in inference, but that is not the trigger for the central limit theorem. Option D is incorrect because the classic central limit theorem for means concerns quantitative data, not purely categorical labels. The theorem is foundational because it justifies using normal-based confidence intervals and hypothesis tests for means when sample sizes are large. Study Guide references/topics: central limit theorem, sample size, sampling distribution, normal approximation.
NEW QUESTION # 128
Confidence interval = sample mean ± ?
Answer: B
Explanation:
A confidence interval for a population mean is generally constructed as sample mean ± margin of error. The margin of error equals a critical value multiplied by the standard error. Therefore, the structure is sample mean ± critical value × SE. The critical value depends on the confidence level and distribution used, such as a z critical value when the population standard deviation is known or a t critical value when it is unknown. The standard error measures the sampling variability of the sample mean, often computed as s/#n when using sample standard deviation. Option B is incorrect because variance times sample size is not a margin of error.
Option C is incomplete because standard deviation alone measures spread among data values, not uncertainty of the sample mean. Option D is incorrect because the median is a measure of center and is not the typical component of a mean-based confidence interval. Study Guide references/topics: confidence intervals, standard error, critical value, margin of error.
NEW QUESTION # 129
Determine x:
4 + x = 6 # x
Answer: A
Explanation:
The equation 4 + x = 6 # x is a linear equation in one variable. To solve it, collect variable terms on one side and constant terms on the other. Add x to both sides to eliminate the negative x on the right: 4 + 2x = 6. Then subtract 4 from both sides: 2x = 2. Finally, divide both sides by 2, giving x = 1. This value can be verified directly in the original equation. Substituting x = 1 gives 4 + 1 = 5 on the left and 6 # 1 = 5 on the right, so both sides are equal. Option A gives 4 = 6, which is false. Option C gives 6 = 4, also false. Option D gives 9 =
1, which is not valid. The correct solution is the value that balances both sides of the equation. References
/topics from the Study Guide: basic algebra, solving linear equations, inverse operations, substitution checks.
NEW QUESTION # 130
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