Data: {2, 4, 6}. What is the variance (population)? โ 4.Squared deviations: 4, 0, 4. Sum = 8. Variance = 8/3 โ 2.67. Wait: 8/3 โ 2.67, not 4. The answer 2.67 is correct.
Data: {2, 4, 6}. What is the population standard deviation? โ โ(8/3) โ 1.63.ฯ = โ(variance) = โ(8/3) โ 1.633.
Which standard deviation is used when data is a SAMPLE (not the entire population)? โ s (divides by n โ 1).Sample standard deviation s uses n โ 1 (Bessel's correction) to provide an unbiased estimate of the population standard deviation.
Two data sets have the same mean but different standard deviations. What does this tell you? โ They have the same center but different spreads.Same mean = same center. Different ฯ = different spread (variability). The sets are centered the same but spread differently.
A dataset has ฯ = 0. What can you conclude? โ All data values are equal (same as the mean).ฯ = 0 means no spread โ all values equal the mean (and each other). The mean can be any value.
For data {6, 7, 8, 9, 10}: mean = 8. Sum of squared deviations = ? โ 10.(6โ8)ยฒ+(7โ8)ยฒ+(8โ8)ยฒ+(9โ8)ยฒ+(10โ8)ยฒ = 4+1+0+1+4 = 10.
For the previous data ({6,7,8,9,10}): population variance = ? โ 2.Variance = sum of squared deviations / n = 10/5 = 2.
For the same data: population ฯ = ? โ โ2.ฯ = โvariance = โ2 โ 1.414.
A factory measures screw lengths. Dataset A has ฯ = 0.01 mm; Dataset B has ฯ = 2 mm. Which factory produces more consistent screws? โ Factory A โ smaller ฯ means less variability.Smaller ฯ means data is more tightly clustered โ more consistent. Factory A (ฯ = 0.01) produces far more consistent screws.
Test scores: ฮผ = 75, ฯ = 10. A student scored 90. What is their z-score? โ 1.5.z = (90 โ 75)/10 = 15/10 = 1.5. This student is 1.5 standard deviations above the mean.
A z-score of โ2 means: โ The value is 2 standard deviations below the mean.Negative z-score means the value is below the mean. z = โ2 means 2 standard deviations below the mean.
Heights: ฮผ = 65 in, ฯ = 4 in. Using the Empirical Rule, about 95% of heights fall between: โ 57 and 73 in.95% falls within ฮผ ยฑ 2ฯ = 65 ยฑ 8 = 57 to 73 inches.
If z = (x โ ฮผ)/ฯ = 1.5, and ฮผ = 50, ฯ = 4, what is x? โ 56.1.5 = (x โ 50)/4 โ x โ 50 = 6 โ x = 56.
In a normal distribution, what is the relationship between mean, median, and mode? โ Mean = Median = Mode.For a normal distribution, the mean, median, and mode are all equal, located at the center of the symmetric bell curve.
A normal distribution with ฮผ = 0 and ฯ = 1 is called the: โ Standard normal distribution.The standard normal distribution has ฮผ = 0 and ฯ = 1. All normal distributions can be standardized to it using z-scores.
Data values with z-scores between โ3 and 3 represent about what percent of normally distributed data? โ 99.7%.The Empirical Rule: 99.7% of data falls within ฮผ ยฑ 3ฯ (z between โ3 and 3).
If ฯ is doubled while keeping ฮผ constant, the bell curve becomes: โ Wider and flatter.Larger ฯ = more spread = the bell curve is wider and flatter. The total area remains 1.
SAT scores: ฮผ = 1000, ฯ = 200. A score of 800 has z-score: โ โ1.z = (800 โ 1000)/200 = โ200/200 = โ1. One standard deviation below the mean.
A z-score of โ0.5 means the value is: โ Below the mean by 0.5 standard deviations.z = โ0.5 means the value is half a standard deviation below the mean. The sign shows direction; magnitude shows distance in ฯ units.
The area under the entire normal curve equals: โ 1.The total area under any probability distribution curve = 1 (representing 100% probability). The normal distribution integrates to 1.
IQ scores have ฮผ = 100, ฯ = 15. Using the Empirical Rule, approximately 99.7% of IQs fall between: โ 55 and 145.ฮผ ยฑ 3ฯ = 100 ยฑ 45 = 55 to 145.
The regression line ลท = 3x + 5 predicts y = 11 for x = 2. The actual y is 9. What is the residual? โ โ2.Residual = actual โ predicted = 9 โ 11 = โ2. Negative residual means the actual value is below the line.
ลท = 1.5x โ 3. What does the y-intercept โ3 represent? โ Predicted y when x = 0.The y-intercept gives the predicted y value when x = 0. Here, the model predicts y = โ3 when x = 0.
The slope of the regression line represents: โ The predicted change in y for each 1-unit increase in x.Slope = predicted change in y per 1-unit increase in x. It quantifies the rate of change in the relationship.
If r = 0.6 and the regression gives a good fit, what type of association exists? โ Moderate positive linear association.|r| = 0.6 is moderate (between 0.5 and 0.8). Positive r means positive direction. Moderate positive linear association.
A residual plot with a random scatter (no pattern) indicates: โ The linear model is APPROPRIATE.Random scatter in a residual plot means the linear model captures the trend well. Any curved pattern suggests a non-linear model is better.
A residual plot showing a curved pattern suggests: โ A non-linear model may be more appropriate.A curved residual pattern suggests the underlying relationship is nonlinear โ a linear model is missing systematic patterns.
ลท = 2x + 1 with r = 0.95. Which interpretation is most accurate? โ There is a strong positive linear relationship and y increases about 2 units per unit of x.Strong r = 0.95 shows a strong positive linear relationship. Slope = 2 means each unit increase in x predicts ~2 unit increase in y on average.
What range of r indicates a WEAK linear association? โ |r| < 0.5.Generally: |r| < 0.5 = weak, 0.5โ0.8 = moderate, |r| > 0.8 = strong. These are rough guidelines.