The use of social networks, for example, Facebook and Twitter, has grown dramati

Question

The use of social networks, for example, Facebook and Twitter, has grown dramatically all over the world. In a recent survey among social network users, it was reported that Indonesians and Saudi Arabians spend the most time per day using social networks. American social network users reportedly spend over 3 hours per day social networking from a computer, tablet, and/or mobile phone. A random sample of 24 American social network users was obtained and each was asked for the amount of time spent (in hours) social networking each day. The sample mean is 3.19 hours with a standard deviation of 0.2903 hours.

A)The sample mean of 3.19 hours is an estimate of the population mean time that Americans spend social networking (in hours) per day. Report the standard error of the sample mean (include your units).

B)Provide a 90% confidence interval estimate for µ, the population mean number of hours that Americans spend social networking per day. Report to two decimal places and include units. Note: you can assume that the distribution of the number of hours that Americans spend social networking per day is approximately normal.

C)Provide an appropriate interpretation of the 90% confidence interval in context.

D)Which of the following interpretations regarding the 90% confidence level are correct? Select all that are correct.

1-If the procedure is repeated many times, we would expect about 90% of the resulting confidence intervals to contain the sample mean number of hours that Americans spend social networking per day.
2- If the procedure is repeated many times, we would expect about 90% of the resulting confidence intervals to contain the population mean number of hours that Americans spend social networking per day.
3-If the procedure is repeated many times, 90% of the time the interval computed in the previous part will contain the population mean number of hours that Americans spend social networking per day.
4- If the procedure is repeated many times, then we’d expect the population mean number of hours that Americans spend social networking per day to fall in 90% of the resulting 90% confidence intervals.

Explanation / Answer

a)

SE = sigma/sqrt(n)

= 0.2903/sqrt(24)

= 0.059257239 [ANSWER]

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b)

Note that              
Margin of Error E = t(alpha/2) * s / sqrt(n)              
Lower Bound = X - t(alpha/2) * s / sqrt(n)              
Upper Bound = X + t(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.05          
X = sample mean =    3.19          
t(alpha/2) = critical t for the confidence interval =    1.713871528          
s = sample standard deviation =    0.2903          
n = sample size =    24          
df = n - 1 =    23          
Thus,              
Margin of Error E =    0.101559295          
Lower bound =    3.088440705          
Upper bound =    3.291559295          
              
Thus, the confidence interval is              
              
(   3.088440705   ,   3.291559295   ) [ANSWER]

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c)

We are 90% confident that the true mean amount of time spent (in hours) social networking each day is between 3.088440705 and 3.291559295 hours. [ANSWER]

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d)

OPTION 2:

2- If the procedure is repeated many times, we would expect about 90% of the resulting confidence intervals to contain the population mean number of hours that Americans spend social networking per day.

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