It seems like you do not have a normal distribution, since the confidence interval goes from .003 below to .013 above .50. The confidence interval is determined by the standard error of the mean (SEm).
SEm = SD/√n
From your data, you would have an extreme large n.
However, without a normal distribution and more data, I cannot answer your last two questions. What was the sample mean? Do you have a typo?
2. A statistics professor has asked his students to flip coins over the years. He has kept track of how many flips land heads and how many land tails. Combining the results of his students over many years, he has formed a 95% confidence interval for the long-run population proportion of heads to be (.497, .513).
a. Why is this interval so narrow?
b. Suppose he were to conduct a hypothesis test of whether the long-run population proportion of heads differs from one-half. Based on this interval (do not conduct the test), would he reject the null hypothesis at the .05 significance level? Explain briefly (no more than one sentence).
c. Does the interval provide strong evidence that the long-run population proportion of heads is much different from one-half? Explain briefly.
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