Test 9 for AP Stats.

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Question 1 |

### What should be used to compute a confidence interval for the slope of a regression model?

A t-interval | |

A z-score | |

A p-value | |

A Chi-squared test |

Question 1 Explanation:

The appropriate confidence interval for the slope of a regression model is a t-interval for the slope. Answer A is correct.

Question 2 |

### Which of the following is not a condition to calculate confidence intervals for the slope of a regression model?

The relationship between x and y is linear. | |

The standard deviation for y varies with x. | |

Data is collected with a random sample. | |

The y-values are normally distributed or $n>30$. |

Question 2 Explanation:

The following conditions must be checked to calculate confidence intervals for the slope of a regression line: the true relationship between x and y is linear, the standard deviation for y does not vary with x (all x should have approximately equal standard deviations), data should be collected with a random sample or randomized experiment and if data is collected without replacement then $n≤10%n$, and the y-values are normally distributed (if the distribution is skewed, n should be greater than 30). Answer B is not a condition.

Question 3 |

### The slope of a given regression line of a sample size of 30 is 1.5, with a sample standard deviation s of 0.3 and a sample standard deviation of the x-values of 0.25. What is the margin of error for a critical value of 2.467?

0.382 | |

0.540 | |

0.550 | |

0.559 |

Question 3 Explanation:

The margin of error is t*SE, the standard error, which can be calculated by

$ \dfrac{s}{s_x\sqrt{n-1}} = \dfrac{0.3}{0.25\sqrt{29}} ≈ 0.223$

$2.467*0.223≈0.550$

Answer C is correct.

$ \dfrac{s}{s_x\sqrt{n-1}} = \dfrac{0.3}{0.25\sqrt{29}} ≈ 0.223$

$2.467*0.223≈0.550$

Answer C is correct.

Question 4 |

### Which of these is the correct calculation of the confidence interval for the slope of the least squares regression line with a confidence interval of 95%, a sample size of 24, a slope of 0.386, and a standard error of sample slope of 0.082? Use the excerpt from the table below to find t*.

## df |
## Tail Probability p=0.05 |
## Tail Probability p=0.025 |

## 22 |
## 1.717 |
## 2.074 |

## 23 |
## 1.714 |
## 2.069 |

## 24 |
## 1.711 |
## 2.064 |

## 25 |
## 1.708 |
## 2.060 |

## 26 |
## 1.706 |
## 2.056 |

$0.386土2.074(0.082)$ | |

$0.386土2.064(0.082)$ | |

$0.386土2.056(0.082)$ | |

$0.082土2.074(0.386)$ |

Question 4 Explanation:

The interval estimate is $b土t*(SE_b)$. In this case, $b=0.386, SE_b=0.082$, and $t* = 2.074 (p=0.025$ with $df n-2=24-2=22)$. The correct set-up for the calculation is $0.386土2.074(0.082)$ and answer A is correct.

Question 5 |

### Which of the following describes the effects of sample size on the width of a confidence interval for the slope of a regression model?

As sample size increases, the width of the confidence interval increases. | |

As sample size increases, the width of the confidence interval decreases. | |

As sample size increases, the width of the confidence interval remains constant. | |

As sample size increases, the width of the confidence interval changes at random. |

Question 5 Explanation:

All other factors remaining the same, the width of the confidence interval will decrease as the sample size increases. Answer B is correct.

Question 6 |

### Which of the following would not be an appropriate null and alternative hypothesis for the slope of a regression model?

$H_0: β=β_0, H_a: β≠β_0$ | |

$H_0: β=β_0, H_a: β<β_0$ | |

$H_0: β=β0_, H_a: β>β_0$ | |

$H_0: β=β_0, H_a: β≤β_0$ |

Question 6 Explanation:

Appropriate alternative hypotheses can use <, >, or ≠ but cannot use ≤ or ≥ as this would overlap with the null hypothesis. Answer D is correct.

Question 7 |

### Which of the following is a condition for the significance test for the slope of a regression model?

The true relationship between x and y is nonlinear. | |

The standard deviation for y does not vary with x. | |

For sampling without replacement, $n≤10%N$ | |

If the distribution of y-values is skewed, n should be at least 20. |

Question 7 Explanation:

Conditions for the significant test include that the true relationship between x and y is nonlinear, the standard deviation for y does not vary with x, $n≤10%N$ for sampling without replacement, and $n>30$ if the distribution of y-values is skewed. Answer C is correct.

Question 8 |

### What is the distribution of the slope of a regression model, assuming all conditions are satisfied and the null hypothesis is true?

Normal distribution | |

Z-distribution | |

T-distribution | |

Chi-squared distribution |

Question 8 Explanation:

The distribution of the slope of a regression model, assuming all conditions are satisfied and the null hypothesis is true, is a t-distribution. Answer C is correct.

Question 9 |

### How should one use the p-value to justify a claim based on the results of a significance test for the slope of a regression model?

If $p-value = a$, accept the null hypothesis. | |

If $p-value ≥ a$, reject the null hypothesis. | |

If $p-value ≤ a$, reject the null hypothesis. | |

If $p-value ≠ a$, accept the null hypothesis. |

Question 9 Explanation:

If the $p-value ≤ a$, then reject the null hypothesis; if $p-value>a$, accept the null hypothesis. Answer C is correct.

Question 10 |

### A student took a random sample of other students and found a linear relationship between the number of minutes spent in the library and ACT scores. A 95% confidence level for the slope of the regression line was (15, 55). The student wants to use this interval to test $H_0: β=β_0, H_a: β≠β_0$ at the α=0.05 level of significance. What is an appropriate conclusion?

Fail to reject $H_0$, and fail to conclude a linear relationship between library use and ACT scores. | |

Fail to reject $H_0$, suggesting a linear relationship between library use and ACT scores. | |

Reject $H_0$, and fail to conclude a linear relationship between library use and ACT scores. | |

Reject $H_0$, suggesting a linear relationship between library use and ACT scores. |

Question 10 Explanation:

The null hypothesis says there is no relationship, that the slope of the regression line = 0. Because the confidence interval does not contain zero, the student can reject the null hypothesis and conclude that there is a linear relationship between library use and ACT scores. Answer D is correct.

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