Question

Suppose that 39% of all U.S. employees contribute to a retirement plan (p = 0.39). (a) In a random sample of 100 employees, what is the approximate probability that at least half of those in the sample contribute to a retirement plan? (Round your answer to four decimal places.) (b) Suppose you were told that at least 71 of the 100 employees in a sample from your state contribute to a retirement plan. Would you think p = 0.39 for your state? Explain. (Round your answer to four decimal places.) If p = 0.39, the probability that at least 71 of the 100 employees in a sample from the state contribute to a retirement plan is . Thus, it seems that p = 0.39 for the state.

Answer 1

The probability that at least half of a sample of 100 employees contribute to a retirement plan, with an underlying rate of 0.39, can be approximated using the normal distribution. Observing that 71 out of 100 employees contribute suggests that the true proportion may be higher than 0.39.

Step-by-step explanation:

To answer part (a) of the question, we need to calculate the probability of at least half (50 out of 100) employees contributing to a retirement plan when given a sample proportion (p) of 0.39. The normal approximation to the binomial distribution can be used since the sample size is large (n=100). While the exact binomial probability could be used, it is complex and the normal approximation is suitable in this case.

For part (b), the observation that at least 71 out of 100 employees contribute to a retirement plan seems much higher than what would be expected if the true proportion were 0.39. To quantify this, a hypothesis test could be used to compare the observed proportion to the hypothesized proportion of 0.39.

The probability of at least 71 of the employees contributing, if p really were 0.39, can be found by calculating the binomial probability or by a normal approximation, but either calculation is expected to show that this is a very rare event. Thus, it would lead us to be skeptical of whether p = 0.39 is true for that state.