According to a 2017 Pew Research Center report on voting...

According to a 2017 Pew Research Center report on voting issues, $59 \%$ of Americans feel that the everything should be done to make it easy for every citizen to vote. Suppose a random sample of 200 Americans is selected. We are interested in finding the probability that the proportion of the sample who feel with way is greater than $55 \%$. a. Without doing any calculations, determine whether this probability will be greater than $50 \%$ or less than $50 \%$. Explain your reasoning. b. Calculate the probability that the sample proportion is $55 \%$ or more.

According to a 2017 Pew Research Center report on voting issues, $59 \%$ of Americans feel that the everything should be done to make it easy for every citizen to vote. Suppose a random sample of 200 Americans is selected. We are interested in finding the probability that the proportion of the sample who feel with way is greater than $55 \%$.
a. Without doing any calculations, determine whether this probability will be greater than $50 \%$ or less than $50 \%$. Explain your reasoning.
b. Calculate the probability that the sample proportion is $55 \%$ or more.

Key Concepts

Standardization (Z-score)

Standardization involves converting a raw statistic into a z-score by subtracting the mean of the sampling distribution and dividing by the standard error. This transformation allows the use of the standard normal distribution to compute probabilities associated with the statistic.

Sampling Distribution of the Sample Proportion

This concept refers to the probability distribution that arises when a sample proportion is computed from many different samples taken from the same population. The distribution has a mean equal to the true population proportion and a standard deviation that quantifies the variability from one sample to another.

Central Limit Theorem

The Central Limit Theorem is a fundamental principle in statistics stating that, for large enough sample sizes, the sampling distribution of almost any statistic (including the sample proportion) will be approximately normally distributed, regardless of the original distribution of the population.

Standard Error of the Sample Proportion

The standard error measures the variability of the sample proportion as an estimator of the true population proportion. It is calculated as the square root of [p(1-p)/n], where p is the population proportion and n is the sample size, quantifying the expected deviation from the true proportion due to sampling variability.

Probability Calculation Using Normal Distribution

Once a statistic is standardized, probabilities can be calculated using the standard normal distribution. This process involves finding the area under the normal curve beyond a specified z-score, which corresponds to the probability of observing a value at least as extreme as the one given.

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