Renaissance Resistors made a random check of 50 of their...

Renaissance Resistors made a random check of 50 of their professional employees who had been provided with either a personal computer or a computer workstation and found that 22 of them used their computers for more than 6 hours per day. (a) Find the maximum likelihood estimate of the true proportion of professionals who use their computers at least 6 hours per day. (b) Find the $95 \%$ confidence interval for the estimate of part (a). (c) How large should the random sample be to ensure that the width of the $95 \%$ confidence interval of part (b) does not exceed 0.1 ?

Renaissance Resistors made a random check of 50 of their professional employees who had been provided with either a personal computer or a computer workstation and found that 22 of them used their computers for more than 6 hours per day.
(a) Find the maximum likelihood estimate of the true proportion of professionals who use their computers at least 6 hours per day.
(b) Find the $95 \%$ confidence interval for the estimate of part (a).
(c) How large should the random sample be to ensure that the width of the $95 \%$ confidence interval of part (b) does not exceed 0.1 ?

Key Concepts

Confidence Interval for a Proportion

A confidence interval for a proportion is a range of values that is believed, with a certain level of confidence (e.g., 95%), to contain the true proportion of successes in the population. It accounts for the sampling variability and is often constructed using the normal approximation to the binomial distribution when the sample size is sufficiently large. The interval typically centers around the sample proportion and extends out by the margin of error calculated using a critical value from the standard normal distribution.

Sample Size Determination in Estimation

Sample size determination in estimation involves calculating the number of observations required to achieve a desired margin of error for a confidence interval. For proportions, this calculation often uses the formula based on the standard normal distribution and incorporates an estimate of the proportion, the critical value corresponding to the desired confidence level, and the acceptable margin of error. This ensures that the confidence interval's width does not exceed a pre-specified length, balancing precision with practicality.

Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) is a method used to estimate the parameters of a statistical model. In the context of a binomial distribution, such as estimating a proportion, the MLE of the true proportion is obtained by the observed proportion of successes in the sample. This method maximizes the likelihood function, which represents the probability of observing the given data under different parameter values.

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