Asample size of 500 is sufficiently large enough to conclude that the sampling distribution of the sample proportions is a normal distribution, when the estimate of the population proportion is .995.

In a statistical test, the null hypothesis to be made is that the sample proportions do not have any significant differences, which means an equal distribution. This is only rejected when the estimate is equal or less than 0.95. But since in this case it is >0.95, so therefore the null hypothesis is not rejected. Therefore:

False

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.