Question

Smartphone users around the world are increasing the frequency of their communications with each other but, most of the time, it's not in the form of a phone call. Based on Prnewswire.com's report, 400 customers (age 18-35) were selected randomly to study the mean number of text messages sent per day. The 400 people in the study sent 10,320 text messages on the day the study was conducted. The standard deviation in the number of texts sent per day by the people in this study was 9.5 messages. a) What is the point estimate for the mean number of text messages sent per day by customers in this age group? b) Calculate and interpret the 95% confidence interval estimate for the mean number of text messages sent per day by customers in this age group. c) If the company would like to have a smaller margin of error, indicate what options are available. d) Based on the result in part b, what conclusion might you reach about the PR Newswire statement about the mean being 32 messages per person?

Answer 1

a) The point estimate for the mean number of text messages sent per day by customers in this age group can be calculated by dividing the total number of text messages sent by the number of customers in the study.

Point estimate = Total number of text messages / Number of customers

Point estimate = 10,320 / 400

Point estimate ≈ 25.8 text messages per day

Therefore, the point estimate for the mean number of text messages sent per day by customers in this age group is approximately 25.8.

b) To calculate the 95% confidence interval estimate for the mean number of text messages sent per day, we can use the formula:

Confidence interval = Point estimate ± (Z * Standard deviation / √n)

Where:

Z is the z-score corresponding to the desired confidence level (95% confidence level corresponds to a z-score of approximately 1.96).

Standard deviation is the standard deviation of the sample (9.5 messages).

n is the sample size (400 customers).

Confidence interval = 25.8 ± (1.96 * 9.5 / √400)

Confidence interval = 25.8 ± (1.96 * 9.5 / 20)

Confidence interval = 25.8 ± 1.84

Confidence interval ≈ (24.0, 27.6)

Interpretation: We can be 95% confident that the true mean number of text messages sent per day by customers in this age group falls within the range of approximately 24.0 to 27.6.

c) To have a smaller margin of error, the company can consider increasing the sample size. By increasing the sample size, the standard error decreases, resulting in a smaller margin of error and a more precise estimate of the population mean.

d) Based on the 95% confidence interval estimate obtained in part b, we can conclude that the PR Newswire statement about the mean number of text messages being 32 messages per person is not supported by the data. The confidence interval does not include the value of 32. Therefore, there is not enough evidence to support the claim that the mean number of text messages sent per day by customers in this age group is 32 messages.