Suppose that 44 % of adult Australians believe that...

Suppose that $44 \%$ of adult Australians believe that Australia should become a republic. Calculate the probability that more than $50 \%$ of a random sample of 100 adult Australians would believe this.

Key Concepts

Binomial Distribution

This is a fundamental concept in probability where the number of successes in a fixed number of independent trials is counted, with each trial having only two possible outcomes (success or failure) and a constant probability of success. It provides the framework for modeling scenarios where we are interested in the probability of a certain number of successes occurring in a series of trials.

Central Limit Theorem

This theorem states that, as the sample size grows, the distribution of the sample proportion (or mean) will tend to be normally distributed, regardless of the shape of the underlying population distribution. This principle justifies using normal approximation for the sampling distribution even when the original variable is binary, provided the sample size is large enough.

Normal Approximation to the Binomial Distribution

When the sample size is large, the binomial distribution can be approximated by a normal distribution, simplifying calculations. This approximation involves using the mean and standard deviation of the binomial distribution (often referred to as the standard error when discussing proportions) to convert the problem into finding a corresponding z-score and using standard normal probability tables.

Standard Error

This is a measure of the variability or spread of the sampling distribution of a statistic, such as the sample proportion. In the context of a binomial distribution, it is calculated as the square root of the product of the probability of success, the probability of failure, and the reciprocal of the sample size, and it quantifies how much the sample proportion is expected to vary from the true population proportion.

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