An app developer seeks to decide between a green and a blue version of a retail app. The developer releases a beta version of the app, and a user who downloads the app is randomly given the green or the blue version. A week after the download, the average time spent on the retail app by the 105 individuals who downloaded the green version and the 103 individuals who downloaded the blue version were, respectively, 14 minutes (SD = 3.3 minutes) and 11 minutes (SD = 3.1 minutes).

1. What is a 80% confidence interval for the difference between the average time spent by users on the green version of the app and the average time spent by users on the blue version of the app?

2. Consider the following interpretations of this 80% confidence interval, Which of them is correct?

(a) We can be 80% confident that the difference between the average time spent by users on the green and on the blue version of the app, in the population, is within the interval obtained.

(b) We can be 80% confident that the difference between the average time spent by users on the green and on the blue version of the app, in the sample, is within the interval obtained.

(c) There is a .80 probability that the difference between the average time spent by users on the green and on the blue version of the app, in the sample, lies in the interval obtained.

(d) There is a .80 probability that the difference between the average time spent by users on the green and on the blue version of the app, in the population, lies in the interval obtained.

3. The 80% confidence interval included only positive values. Can we be sure that the 99% confidence interval also includes only positive values?

4. What is an implicit assumption that this analysis needs to rely on?

5. Considering the way that this analysis was carried out, which of the following should be said about the population to which the findings can generalize?

(a) The population is all individuals who buy the type of products sold on the retail app.

(b) The population is all individuals with a smartphone.

(c) We should be careful to extend the findings to all potential users of this retail app, because the sample only included individuals who voluntarily downloaded the beta version of the app.

Community Answer

Honor CodeSolved 1 Answer

See More Answers for FREE

Enhance your learning with StudyX

Receive support from our dedicated community users and experts

See up to 20 answers per week for free

Experience reliable customer service

Get Started

We need to construct the 80%  confidence interval for the difference between the population means  \mu_{1}-\mu_{2}​, for the case that the population standard deviations are not known. The following information has been provided about each of the samples: Sample Mean 1  \left(\bar{X}_{1}\right) = 14 Sample Standard Deviation 1 (s1)  = 3.3 Sample Size 1 (N1)  = 105 Sample Mean 2  \left(\bar{X}_{2}\right)= 11 Sample Standard Deviation 2 (s2)  = 3.1 Sample Size 2 (N2)  = 103 Based on the information provided, we assume that the population variances are equal, so then the number of degrees of freedom are df=n1+n2−2=105+103−2=206. The critical value for  \alpha=0.2 and df=206 degrees of freedom is  t_{c}=t_{1-\alpha / 2 ; n-1}=1.286. The corresponding confidence interval is computed as shown below: Since the population variances are assumed to be equal, we need to compute the pooled standard deviation, as follows:  \begin{array}{c}s_{p}=\sqrt{\frac{\left(n_{1}-1\right) s_{1}^{2}+\left(n_{2}-1\right) s_{2}^{2}}{n_{1}+n_{2}-2}} \\ =\sqrt{\frac{(105-1) \times 3.3^{2}+(103-1) \times 3.1^{2}}{105+103-2}} \\ =3.203\end{array} Since we assume that the population variances are equal, the standard error is computed as follows:  \begin{array}{c}s e=s_{p} \sqrt{\frac{1}{n_{1}}+\frac{1}{n_{2}}} \\ =3.203 \times \sqrt{\frac{1}{105}+\frac{1}{103}} \\ =0.444\end{array} Now, we finally compute the confidence interval:  \begin{array}{c}C I=\left(X_{1}-X_{2}-t_{c} \times s e, X_{1}-X_{2}+t_{c} \times s e\right) \\ =(14-11-1.286 \times 0.444,14-11+1.286 \times 0.444) \\ =(2.429,3.571)\end{array} Therefore, based on the data provided, the 80%  confidence interval for the difference between the population means \mu_{1}-\mu_{2}​ is  2.429<\mu_{1}-\mu_{2}<3.571, which indicates that we are 80%  confident that the true difference between population means is contained by the interval (2.429,3.571).   (2) option a is correct. (a) We can be 80% confident that the difference between the average time spent by users on the green and on the blue version of the app, in the population, is within the interval obtained. (4)   What is an implicit assumption that this analysis needs to rely on? option 2 is correct.  that the first users to download the beta version of the app can be regarded as the random sample from the population.       ...