CONFIDENCE INTERVAL AND HYPOTHESIS TESTING

CONFIDENCE INTERVAL AND HYPOTHESIS TESTING


C
Asked by 1 year ago
10 million points

5). Suppose a consumer advocacy group would like to conduct a survey to find the proportion p of consumers who bought the newest generation of an MP3 player were happy with their purchase.

5a) How large a sample $n$ should they take to estimate p with $2\%$ margin of error and $90\%$ confidence?

5b) The advocacy group took a random sample of $1000$ consumers who recently purchased this MP3 player and found that $400$ were happy with their purchase. Find a $95\%$ confidence interval for p.

6). In order to ensure efficient usage of a server, it is necessary to estimate the mean number of concurrent users. According to records, the sample mean and sample standard deviation of number of concurrent users at $100$ randomly selected times is $37.7$ and $9.2$, respectively.

6a) Construct a $90\%$ confidence interval for the mean number of concurrent users.

6b) Do these data provide significant evidence, at $1\%$ significance level, that the mean number of concurrent users is greater than $35$?

7). To assess the accuracy of a laboratory scale, a standard weight that is known to weigh $1$ gram is repeatedly weighed $4$ times. The resulting measurements (in grams) are: $0.95, 1.02, 1.01, 0.98$. Assume that the weighing's by the scale when the true weight is $1$ gram are normally distributed with mean $\mu$.

7a) Use these data to compute a $95\%$ confidence interval for $\mu$.

7b) Do these data give evidence at $5\%$ significance level that the scale is not accurate? Answer this question by performing an appropriate test of the hypothesis.

CONFIDENCE INTERVAL
HYPOTHESIS TESTING

2 Answers

C
Answered by 1 year ago
10 million points

SOLUTION

5)

a) It is given that margin of error $E = 0.02$ and $\alpha = 0.10$. Using $p = 0.5$ as the conservative guess in the sample size formula gives,

$$n =\bigg[p(1−p)\bigg(\frac{Z_{\frac{\alpha}{2}}}{E}\bigg)^{2}\bigg]=\bigg[\bigg(\frac{Z_{\frac{\alpha}{2}}}{2E}\bigg)^{2}\bigg]=\bigg(\frac{1.645}{0.04}\bigg)^2= 1692.$$

b) From the data, $\bar{p} = \frac{400}{1000} = 0.40$. Since $n = 1000$ is large, the $90\%$ condence interval for $p$ is:

$$p±Z_{\frac{\alpha}{2}}\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}= 0.40±1.645\sqrt{\frac{0.400.60}{1000}} = [0.375,0.425]. $$

6) Let $\mu$ denote the mean number of concurrent users in the population. It is given that $n = 100, X = 37.7$ and $S = 9.2$.

a) We want a $90\%$ condence interval for $\mu$. Since $n$ is large, we will use the large sample $Z$-interval: $$\bigg(X ±Z_{\frac{\alpha}{2}} \frac{S}{√n}\bigg)$$.

From the normal table, $Z_{0.05} = 1.645$. Thus, the desired condence interval is:

$$37.7±(1.645)\frac{9.2}{√100} = 37.7±1.5 = [36.2,39.2]$$.

b) We need to test the null $H_{0} : \mu = 35$ against the one-sided alternative $H_{1} : \mu > 35$, at level $\alpha = 0.01$. Since $n$ is large, we will do a large-sample $Z$-test. The rejection region is $Z > Z_{\alpha} = 2.33$, using the normal table. The test statistic is

$$Z =\frac{\bar{X} −\mu_{0}} {\frac{S}{√n}} =\frac{37.7-35}{\frac{9.2}{√100}}=2.93$$

Since $Z = 2.93 > 2.33$, $H_0$ is rejected. Thus, there is signicant evidence at $1\%$ signicance level that the mean number of concurrent users is greater than $35$.

7)

a) From the given data, we have: $n = 4, X = 0.99$ and $S = 0.032$. Since $n$ is small and the data are normally distributed we will use the t-interval:

$$X ±t_{n−1,\frac{\alpha}{2}}\frac{S}{√n}=0.99±3.182\frac{0.032}{2}= 0.99±0.05 = [0.94,1.04].$$

b) We need to test the null hypothesis $H_{0} : \mu = 1$ against the two-sided alternative $H_{1} : \mu \neq 1$. Since the null value of $\mu = 1$ falls in the $95\%$ condence interval computed in the previous part, it follows that the $5\%$ level test of does not reject $H_{0}$. Thus, there is no evidence at $5\%$ signicance level that the scale is inaccurate.

R
Answered by 1 year ago
0 points

Oh Snap! This Answer is Locked

CONFIDENCE INTERVAL AND HYPOTHESIS TESTING

Thumbnail of first page

Excerpt from file: Individual School of Business MGT/434 Employment Law Affirmative Action Paper prepare a 1,050- to 1,750-word paper in which you describe the elements of affirmative action as it applies to public sector and private sector employers and how it interacts with Title VII requirements of Equal

Filename: Affirmative Action Paper wk 4.docx

Filesize: < 2 MB

Downloads: 2

Print Length: 6 Pages/Slides

Words: 244

Your Answer

Surround your text in *italics* or **bold**, to write a math equation use, for example, $x^2+2x+1=0$ or $$\beta^2-1=0$$

Use LaTeX to type formulas and markdown to format text. See example.

Sign up or Log in

  • Answer the question above my logging into the following networks
Sign in
Sign in
Sign in

Post as a guest

  • Your email will not be shared or posted anywhere on our site
  •  

Stats
Views: 289
Asked: 1 year ago

Related