Test 7 for AP Stats.

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Question 1 |

### When is it appropriate to use a one-sample t-interval for a mean?

When the population mean and standard deviation are both known. | |

When neither the population mean nor the standard deviation are known. | |

When the population mean is known but the population standard deviation is not. | |

When the population mean is unknown but the population standard deviation is known. |

Question 1 Explanation:

When population standard deviation is not known, the appropriate confidence interval procedure for estimating the population mean of one quantitative variable for one sample is a one-sample t-interval for a mean. Answer C is correct.

Question 2 |

### Which statements are true of t-distributions?

### I. More of the area of the distribution is allocated to the tails than in a normal distribution.

### II. Less of the area of the distribution is allocated to the tails than in a normal distribution.

### III. As degrees of freedom increase, the area in the tails of a t-distribution increases.

### IV. As degrees of freedom increase, the area in the tails of a t-distribution decreases.

I and III | |

I and IV | |

II and III | |

II and IV |

Question 2 Explanation:

A t-distribution differs from a normal distribution with more of the area allocated to the tails. The area in the tails decreases and degrees of freedom increase. Answer B is correct.

Question 3 |

### What is the margin of error for a one-sample t-interval with sample size 20 and sample standard deviation 1.2 with a confidence level of 95%? Use the condensed table of values below.

## df |
## 95% |

## 19 |
## 2.093 |

## 20 |
## 2.086 |

## 21 |
## 2.080 |

0.5581 | |

0.5597 | |

0.5616 | |

0.5762 |

Question 3 Explanation:

The formula is $ t*(\dfrac{s}{\sqrt{n}}) $ where $t*$ is the value obtained using degrees of freedom $(df=n-1)$ on a chart, s is the sample standard deviation, and n is the sample size. Answer C is correct.

Question 4 |

### Which of the following statements is NOT true about the interpretation of a confidence interval?

We can be C% confident that the confidence interval for a population captures the population mean. | |

An interpretation of the confidence interval includes a reference to the sample and details about the population it represents. | |

Each interval is based on data from a random sample. | |

The confidence interval will be the same for every random sample. |

Question 4 Explanation:

Because the data for every random sample will vary from sample to sample, the confidence interval will not be the same for every random sample. Answer D is correct.

Question 5 |

### Which could be an appropriate null and alternative hypothesis for a one-sample t-test for a population mean?

$H_0:µ=µ_0 ,H_a:µ≠µ_0$ | |

$H_0:µ<µ_0 ,H_a:µ>µ_0$ | |

$H_0:µ≠µ_0 ,H_a:µ=µ_0$ | |

$H_0:µ=µ_0 ,H_a:µ≥µ_0$ |

Question 5 Explanation:

For a one-sample t-test, $H_0 is µ=µ_0$ where $µ_0$ is the hypothesized value. $H_a$ may be $µ<µ_0 , µ>µ_0$, or $µ≠µ_0$. Answer A is correct.

Question 6 |

### What should an interpretation of the p-value of a significance test for a population mean recognize?

The p-value is computed by assuming the true population mean is equal to the value stated in the null hypothesis. | |

The p-value is computed by assuming the true population mean is equal to the value stated in the alternative hypothesis. | |

The p-value is computed by assuming the true population mean is not equal to the value stated in the null hypothesis. | |

The p-value is computed without input from the null or alternative hypothesis. |

Question 6 Explanation:

The p-value is computed by assuming the true population mean is equal to the value stated in the null hypothesis. Because the alternative hypothesis is not a statement of equality, but of inequality, answer B is impossible. Answer A is correct.

Question 7 |

### Which of the following is a necessary condition to calculate confidence intervals for the difference of two population means?

Sampling distribution of $(x_1-x_2)$ should be approximately normal; if skewed, either $n_1$ or $n_2$ must be greater than 30. | |

Sampling distribution of $(x_1-x_2)$ should be approximately normal; if skewed, both $n_1$ and $n_2$ must be greater than 30. | |

Sampling distribution of $(x_1-x_2)$ should be approximately normal; if skewed, $n_1$ and $n_2$ must be equal to 30. | |

None of the above are necessary conditions. |

Question 7 Explanation:

To calculate confidence intervals for the difference of two population means, the sampling distribution should be approximately normal; if the observations are skewed, both $n_1$ and $n_2$ must be greater than 30. Answer B is correct.

Question 8 |

### What effect does the sample size have on the width of a confidence interval for the difference of two means?

The width of the confidence interval for the difference of two means does not change as the sample sizes increase. | |

The width of the confidence interval for the difference of two means tends to increase as the sample sizes increase. | |

The width of the confidence interval for the difference of two means tends to increase exponentially as the sample sizes increase. | |

The width of the confidence interval for the difference of two means tends to decrease as the sample sizes increase. |

Question 8 Explanation:

The width of the confidence interval for the difference of two means tends to decrease as the sample sizes increase, all other factors remaining the same. Answer D is correct.

Question 9 |

### What could not be appropriate null and alternative hypotheses for a two-sample t-test for a difference of two population means?

$H_0:µ_1=µ_2 ,H_a:µ_1≤µ_2$ | |

$H_0:µ_1=µ_2 ,H_a:µ_1>µ_2$ | |

$H_0:µ_1-µ_2=0 ,H_a:µ_1<µ_2$ | |

$H_0:µ_1-µ_2=0 ,H_a:µ_1≠µ_2$ |

Question 9 Explanation:

An appropriate null hypothesis is a statement of inequality, either $µ_1=µ_2$ or $µ_1-µ_2=0$. An appropriate alternative hypothesis can not contain an overlapping statement, so $µ_1≤µ_2$ or $µ_1≥µ_2$ could not be appropriate alternative hypotheses. Answer A is correct.

Question 10 |

### When conducting a test for the difference of two population means, what is the largest possible degree of freedom for sample sizes of 108 and 110.

107 | |

109 | |

216 | |

217 |

Question 10 Explanation:

The highest possible value for degrees of freedom is $n_1+n_2-2. 108+110-2=216$ so the highest possible degrees of freedom is 216. Answer C is correct.

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