Analysis: 1. Does the theoretical probability match the empirical probability? Explain why this is or whether this poses a problem. 2. Fill in the blank: The Law of Large Numbers tells us that the empirical probability approaches the theoretical probability when 3. Explain how to change the Procedure for the Empirical Probability so that the empirical probability more closely resembles the theoretical probability. 4. Implement the change you proposed in the previous problem. Explain the results of this implementation below. (Include screenshots from StatCrunch.)

Just how accurate are the weather forecasts we hear every day? The table below compares the daily forecast with a city's actual weather for a year.   Actual Weather Forecast Rain No rain Rain 34 56 No rain 4 271 What is the probability that there was no rain forecasted or rain actually happened?

PRACTICE ANOTHER   Recall that Benford's Law claims that numbers chosen from very large data files tend to have "1" as the first nonzero digit disproportionately often. In fact, research has shown that if you randomly draw a number from a very large data file, the probability of getting a number with "1" as the leading digit is about 0.301. Now suppose you are the auditor for a very large corporation. The revenue file contains millions of numbers in a large computer data bank. You draw a random sample of n = 225 numbers from this file and r = 85 have a first nonzero digit of 1. Let p represent the population proportion of all numbers in the computer file that have a leading digit of 1.   What is the value of the sample test statistic? (Round your answer to two decimal places.) ANSWER IS NOT 2.57   Find the P-value of the test statistic. (Round your answer to four decimal places.) ANSWER IS NOT 0.0102

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Provide a long step-by-step solution to this problem (see attached). Make your solution with the given and sketch. The solution should be very clear to read, and easy to understand.

Plzz solve a, b, c questions. And explain. Plzz attempt all questions.