Question

Question #3 You want to rent an unfurnished one-bedroom apartment in Boston next year. A 95% confidence interval for the true mean monthly rent of all one-bedroom apartments in Boston is (1263.64, 1536.36). What is the margin of error for this interval?
Question #4 A student curious about the average number of chocolate chips in a commercial brand of cookie estimated the standard deviation to be 8. If he wants to be 99% confident in his results, how manychocolate chip cookies will he need to sample to estimate the mean to within 2?

Answer 1

1.

$$margin\ of\ error={1 \over 2}\times (length\ of\ the\ interval)$$

$$margin\ of\ error={1 \over 2}\times(1536.36-1263.64)$$

$$margin\ of\ error=136.36$$

2.Based on the information provided, the significance level is $\alpha=0.01$ ($99\%$ confidence interval), and the critical value for a two-tailed test is $z_c=2.58$

$$margin\ of\ error=z_c\times{\sigma \over\sqrt{n}}$$

Given that $\sigma=8, margin\ of\ error\leq2.$ Then

$$2.58\times{8 \over\sqrt{n}}\leq2$$

$$n\geq({2.58\times8 \over 2})^2$$

$$n\geq106.5$$

Since $n$ is integer, then $n\geq107.$