Chapter 9, Problem 72CR

The article “Write It by Hand to Make It Stick” (Advertising Age, July 27, 2016) summarizes data from a survey of 1001 students age 13 to 19. Of the students surveyed, 851 reported that they learn best using a mix of digital and nondigital tools. Construct and interpret a 95% confidence interval for the proportion of students age 13 to 19 who would say that they learn best using a mix of digital and non-digital tools. For the method used to construct the interval to be valid, what assumption about the sample must be reasonable?

To determine

Find the 95% confidence interval for the proportion of student’s age 13 to 19 who will say that they learn best using a mix of digital and non-digital tools and interpret it. Also, explain the assumption required for the confidence interval to be valid.

Answer to Problem 72CR

The 95% confidence interval for the proportion of student’s age 13 to 19 who will say that they learn best using a mix of digital and non-digital tools is (0.828,0.872).

Interpretation of confidence interval is “One can be 95% confident that the proportion of student’s age 13 to 19 who will say that they learn best using a mix of digital and non-digital tools is between 0.828 and 0.872”.

Explanation of Solution

Calculation:

It was found that 851 of the 1,001 students surveyed reported that they learn best using a mix of digital and non-digital tools.

The confidence interval for a population proportion p is appropriate in given cases:

• The sample size is large. That is, when $n\widehat{p}\ge 10\text{ and }n\left(1-\widehat{p}\right)\ge 10$,
• The sample size is small relative to the population size. That is, the sample size should not be more than 10% of the population size.
• The sampling method is simple random sampling.

Then, the general formula for finding the confidence interval for a population proportion p is, $\widehat{p}\pm \left(z\text{ critical value}\right)\sqrt{\frac{\widehat{p}\left(1-\widehat{p}\right)}{n}}$ , where $\widehat{p}$ is the sample proportion, and n is the sample size.

The number of student’s age 13 to 19 who will say that they learn best using a mix of digital and non-digital tools is 851 of the 1,001 students surveyed. That is, the sample proportion is, $\frac{851}{1,001}$ .

Check conditions:

Substitute, $n=1,001\text{ and }\widehat{p}=\frac{851}{1,001}$.

$\begin{array}{c}n\widehat{p}=1,001\left(\frac{851}{1,001}\right)\\ =851\\ >10\end{array}$

$\begin{array}{c}n\left(1-\widehat{p}\right)=1,001\left(1-\frac{851}{1,001}\right)\\ =1,001\left(\frac{150}{1,001}\right)\\ =150\\ >10\end{array}$

Since, both the conditions are satisfied and the sample is large, which makes the confidence interval appropriate.

The population is considered for student’s aging 13 to 19 who will say that they learn best using a mix of digital and non-digital tools. A sample of 1,001 students is selected for the survey. Since the population considered is large, the sample size will be smaller than 10% of the population size.

It is assumed that the sample is a representative of the population of student’s age 13 to 19 years. Thus, it is reasonable to consider the sample as a random sample from the population.

The 95% confidence interval for proportion is calculated below:

Software procedure:

Step-by-step procedure to obtain the confidence interval using MINITAB software is given below:

• Choose Stat > Basic statistics>1-Sample proportion.
• In Summarized data, enter Number of events as 851 and Number of trials as 1,001.
• Check Options, enter Confidence level as 95.
• Choose not equal in alternative hypothesis.
• In method select Normal Approximation.
• Click OK in all dialogue boxes.

The output using the MINITAB software is given below:

From the MINITAB output, the confidence interval is (0.827, 0.872).

Thus, the 95% confidence interval for the proportion of student’s age 13 to 19 who will say that they learn best using a mix of digital and non-digital tools is (0.828,0.872).

Interpretation of confidence interval is “it is 95% confident that the proportion of student’s age 13 to 19 who will say that they learn best using a mix of digital and non-digital tools is between 0.828 and 0.872”.

The assumption made for the confidence interval to be valid is that the sample was selected randomly and it is a representative of the population of all students age 13 to 19.