AP Stats and Power

Interpret the circled point on this relative cumulative frequency graph.

• 60% of Starbucks drinks have 30g of sugar.

• 60% of Starbucks drinks have 30g of sugar or more.

• 60% of Starbucks drinks have 30g of sugar or less

• None of these interpretations is correct.

When theory or experience suggests that the relationship between two variables follows a ___________ ___________ of the form y=ax^p, transformations involving powers and roots can linearize a curved pattern in a scatterplot.

Another useful strategy for straightening a curved pattern in a scatterplot is to take the ___________________ of one or both variables. When a power model describes the relationship between two variables, a plot of log y versus log x (or ln y versus ln x) should be linear.

Check all that can be represented by a power model.

• The distance that an object dropped from a given height falls is related to time since release

• The time it takes a pendulum to complete one back-and-forth swing (its period) is related to its length

• The intensity of a light bulb is related to distance from the bulb

When choosing between different models to describe a relationship between two quantitative variables (select all that apply):

• If there is more than one model with a randomly scattered residual plot, choose the model with the largest coefficient of determination, r².

• use the standard deviation of the residuals to help choose a model, as the y values for the different models might be on different scales

• Choose the model whose residual plot has the most random scatter.

A 95% confidence interval for µ based on 𝑛 = 15 observations from a Normal population is (–0.73, 1.92). If we use this confidence interval to test the hypothesis 𝐻₀: 𝜇 = 0 against 𝐻_a: 𝜇 ≠ 0, what is the appropriate conclusion?

​ (a)   H₀ at the α = ​ (b)  

• fail to reject

• reject

• .05

• .10

Answer explanation

Because the hypothesized value, 0, is one of the plausible values in the 95% confidence interval, we fail to reject 𝐻₀:𝜇 = 0 at the α = 0.05 level.

You are thinking of conducting a one-sample 𝑡 test about a population mean 𝜇 using a 0.05 significance level. Which of the following statements is correct?

• You can carry out the test only if the population standard deviation is known.

• You can safely carry out the test if there are no outliers, regardless of the sample size.

• You can safely carry out the test if your sample size is at least 30.

• You can carry out the test if a graph of the data shows no strong skewness, regardless of the sample size.

• You should not carry out the test if the sample does not have a Normal distribution.

Answer explanation

We cannot use 𝑡 procedures if the graph of the sample data shows strong skewness OR outliers.

Which of the following has the smallest probability?

• 𝑃(𝑧>2) if 𝑧 is a standard Normal random variable.

• 𝑃(𝑡>2)if 𝑡 has 2 degrees of freedom.

• 𝑃(𝑧<2) if 𝑧 is a standard Normal random variable.

• 𝑃(𝑡<2) if 𝑡 has 5 degrees of freedom.

• 𝑃(𝑡>2)if 𝑡 has 5 degrees of freedom.

Answer explanation

The standard Normal distribution has less area in the tail than a 𝑡 distribution with df = 2 or df = 5.

A significance test was performed to test 𝐻₀ : 𝜇 = 2 versus the alternative 𝐻_a: 𝜇 ≠ 2. A sample of size 28 produced a standardized test statistic of 𝑡 = 2.051. Assume all conditions for inference are met. Using technology the 𝑃-value is  (Round to 4 decimal places)   ___?

Answer explanation

tcdf(2.051, 1000, 27)

Double the answer to account for both tails of the distribution.

Using Table B and df = 27, 𝑡 = 2.051 falls between 0.025 and 0.05. Because this is a two-sided test, the 𝑃-value falls between 0.05 and 0.10.

A study of road rage asked separate random samples of 596 men and 523 women about their behavior while driving. Based on their answers, each respondent was assigned a road rage score on a scale of 0 to 20. Are the conditions for performing a two-sample 𝑡 test satisfied?

• Maybe; we have independent random samples, but we should look at the data to check Normality.

• Yes; the large sample sizes guarantee that the corresponding population distributions will be Normal.

• No; we don't know the population standard deviations.

• No; road rage scores on a scale from 0 to 20 can't be Normal.

• Yes; we have two independent random samples and large sample sizes.

Answer explanation

Because the values are on a scale of 1 to 20, the populations can’t be Normal. However, the large sample sizes justify the use of the two-sample 𝑡 test. Also, the data come from independent random samples that are both less than 10% of their respective populations.