Question

ch 16* suppose that we took an srs of 1600 eighth - graders and found $x^{-}=288$. compared with an srs of 900 eighth - graders, the margin of error for a 95% confidence interval for $mu$ is
the same.
smaller.
larger.
question 4
1 pts
ch 16 you measure the lifetime of a random sample of 64 tires of a certain brand. in a follow - up study, more tires were available for testing, so you were able to measure the lifetimes of a random sample of 100 tires rather than 64. which of the following statements is true?
the margin of error for our 99% confidence interval would increase.
the margin of error for our 99% confidence interval would stay the same because the level of confidence has not changed.
$sigma$ would decrease.
the margin of error for our 99% confidence interval would decrease.

Explanation:

Step1: Recall margin - of - error formula

The formula for the margin of error $E$ of a confidence interval for the population mean (when population standard deviation $\sigma$ is known) is $E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$, where $z_{\alpha/2}$ is the z - score corresponding to the level of confidence, $\sigma$ is the population standard deviation, and $n$ is the sample size.

Step2: Analyze first problem

For the first problem, comparing an SRS of $n_1 = 1600$ and $n_2=900$ eighth - graders for a 95% confidence interval. Since $z_{\alpha/2}$ and $\sigma$ are assumed to be constant, and $E\propto\frac{1}{\sqrt{n}}$. As $n_1 = 1600>n_2 = 900$, $\sqrt{n_1}>\sqrt{n_2}$, so $\frac{1}{\sqrt{n_1}}<\frac{1}{\sqrt{n_2}}$, and the margin of error for $n = 1600$ is smaller.

Step3: Analyze second problem

For the second problem, comparing a sample of $n_1 = 64$ and $n_2 = 100$ tires for a 99% confidence interval. With $z_{\alpha/2}$ and $\sigma$ assumed constant, and $E\propto\frac{1}{\sqrt{n}}$. Since $n_2=100 > n_1 = 64$, $\sqrt{n_2}>\sqrt{n_1}$, so $\frac{1}{\sqrt{n_2}}<\frac{1}{\sqrt{n_1}}$, and the margin of error for $n = 100$ is smaller, which means the margin of error for $n = 64$ is larger. The statement "The margin of error for our 99% confidence interval would decrease" is true.

Answer:

First question: smaller.
Second question: The margin of error for our 99% confidence interval would decrease.