In a state,52%of the voters supports party-R, and 48% supports party-D. In second state 47% of the voters supports party-D. Suppose 100 voters are surveyed from each state. Assume the survey uses simple random sampling. what is the probability that the survey will show a greater percentage of party-R voters supporting in the second state than in the first state?
To find the probability that the survey will show a greater percentage of party-R voters supporting in the second state than in the first state, we need to calculate the probability of the event happening.
Let's break down the problem and calculate step by step:
Step 1: Determine the sample size for each state
In both states, 100 voters are surveyed. Therefore, the sample size for each state is 100.
Step 2: Calculate the range of votes for party-R in the first state
In the first state, the percentage of party-R voters is 52%. To calculate the range of party-R votes, we find the minimum and maximum possible number of voters supporting party-R:
Min votes for party-R = Percentage * Sample Size = 0.52 * 100 = 52
Max votes for party-R = (100 - Percentage) * Sample Size = 0.48 * 100 = 48
So, in the first state, the number of voters supporting party-R can range from 48 to 52.
Step 3: Calculate the number of votes for party-R required in the second state
To have a greater percentage of party-R voters in the second state than in the first state, we need the number of party-R voters in the second state to be greater than the maximum number of party-R voters in the first state.
Hence, we need more than 52 votes for party-R in the second state.
Step 4: Calculate the probability of having more than 52 votes for party-R in the second state
To calculate the probability, we need to assume a probability distribution. Since the survey uses simple random sampling and no other information is given, we can assume that the number of party-D voters in each state follows a binomial distribution.
To calculate the probability, we will use binomial probability formula:
P(X > 52) = 1 - P(X <= 52)
Where X follows a binomial distribution with parameters (n,p), where n is the sample size and p is the probability of success (percentage of party-D voters).
In the second state, the percentage of party-D voters is 47%. Therefore, the probability of success (p) is 0.47.
Using the binomial probability formula, we can calculate P(X <= 52):
P(X <= 52) = Σ ( nCi * p^i * (1-p)^(n-i) ) for i = 0 to 52
Coming up with the exact probability requires computation, but you can use statistical software or calculators with binomial distribution functions to find the probability.
Remember to subtract this probability from 1 to get the desired probability:
P(X > 52) = 1 - P(X <= 52)
So, the final step is to calculate this probability using binomial distribution calculations.