College statistics

Question

A researcher is interested in finding a 90% confidence interval for the mean number of times per day that college students text. The study included 103 students who averaged 34.9 texts per day. The standard deviation was 19.7 texts. Round answers to 3 decimal places where possible.

a. To compute the confidence interval use a ? z t  distribution.

b. With 90% confidence the population mean number of texts per day is between  and  texts. 

c. If many groups of 103 randomly selected members are studied, then a different confidence interval would be produced from each group. About  percent of these confidence intervals will contain the true population number of texts per day and about  percent will not contain the true population mean number of texts per day. 

Answer 1

a) since population parameters are being estimated from a sample, a t distribution is used

b) The population mean is being questioned so the standard deviation of the mean (σ_x¯ or std error) must be calculated and the t value for (1-α/2) must be found (alpha - 10%) CI = µ ± t^c _* σ_x¯

σ_x¯ = 19.7/ √103. or 1.941. t critical for (1-α/2 ) and 102 deg of freedom [ i.e. t _(.95,102) = 1.660 ]

CI for the number of the mean is then. 34.900 ± 3.222. or [31.678 38.122]

c) If we assume that all these multiple confidence intervals continue to use the same alpha (10%) than we would expect over time that 10% of the confidence intervals from these new samples would NOT contain the true population mean and contrawise, 90% would contain the true population mean. That is what is meant by a 90% confidence interval.