Final answer:
The question revolves around probability and the application of the normal approximation to estimate chances of specific outcomes in coin tosses. Estimations would be done using z-scores and standard normal distribution, but there is likely a typo regarding the mean and standard deviation, which needs clarification for accurate calculations.
Step-by-step explanation:
The subject of this question is probability, specifically calculating the probability of certain outcomes when flipping a coin multiple times. Let's address the calculations involved in estimating the chance of getting exactly 40 heads and getting between 45 and 55 heads inclusive when a coin is tossed 100 times with a given standard deviation (SD) of 0.5 for the box model.
To estimate the chance of getting exactly 40 heads (X=40), one would typically use the binomial probability formula. But since we're asked to estimate, we can use the normal approximation with the continuity correction factor. According to the question, the mean (μ) is provided as 20 and the standard deviation (σ) as 4. However, these values seem inconsistent with a fair coin where p would be 0.5, so there is likely a typo, and we would expect μ to be 50 and σ to be 5 for a fair coin tossed 100 times. Assuming a fair coin, we would calculate the z-score for 40 heads and use standard normal distribution tables or a calculator to find the probability.
For the range of 45 to 55 heads inclusive, the same normal approximation can be used. We would calculate the z-scores for 44.5 heads (45 - 0.5 for continuity correction) and 55.5 (55 + 0.5 for continuity correction) and find the probability for this range.
However, without a proper understanding of the true parameters (mean and standard deviation consistent with a fair coin), this estimation would be proportionally inaccurate. Thus, one must first clarify the proper parameters before proceeding with the calculations.