Suppose that 90 % of all registered California voters favor...

Suppose that $90 \%$ of all registered California voters favor banning the release of information from exit polls in presidential elections until after the polls in California close. A random sample of 25 registered California voters is to be selected. a. What is the probability that more than 20 favor the ban? b. What is the probability that at least 20 favor the ban? c. What are the mean value and standard deviation of the number of voters in the sample who favor the ban? d. If fewer than 20 voters in the sample favor the ban, is this at odds with the assertion that (at least) $90 \%$ of California registered voters favors the ban? (Hint: Consider $P(x<20)$ when $p=.9 .)$

Suppose that $90 \%$ of all registered California voters favor banning the release of information from exit polls in presidential elections until after the polls in California close. A random sample of 25 registered California voters is to be selected.
a. What is the probability that more than 20 favor the ban?
b. What is the probability that at least 20 favor the ban?
c. What are the mean value and standard deviation of the number of voters in the sample who favor the ban?
d. If fewer than 20 voters in the sample favor the ban, is this at odds with the assertion that (at least) $90 \%$ of California registered voters favors the ban? (Hint:
Consider $P(x<20)$ when $p=.9 .)$

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Key Concepts

Mean and Standard Deviation in Binomial Distribution

For a binomial random variable, the mean (expected value) is given by np and the standard deviation by the square root of np(1-p). These provide key measures of central tendency and dispersion in the distribution of the number of successes.

Statistical Inference from Sample Data

This concept involves comparing observed outcomes from a sample with what is expected under a stated population parameter, such as a claimed proportion. It is used to assess whether an observed sample outcome is consistent with the population assertion, potentially highlighting significant deviations.

Binomial Distribution

This is a probability distribution used when there are exactly two possible outcomes (commonly referred to as success and failure) for each of a fixed number of independent trials. The probability of exactly k successes in n trials is computed using the binomial formula, which involves combinations of trials and the constant success probability p and its complement.

Calculation of Binomial Probabilities

This involves summing individual probabilities computed from the binomial probability mass function to obtain the probability of events like 'more than' or 'at least' a certain number of successes. These calculations rely on the application of the binomial formula across a range of values.

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