At 95% confidence, how large a sample should be taken to obtain a margin of error of 0.08 for the estimation of a population proportion? Assume that past data are not available for developing a planning value for p*. (Round your answer up to the nearest whole number.) 150
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
![**Sample Size Calculation for Population Proportion**
When planning a study to estimate a population proportion with a specific level of precision, it's crucial to determine the appropriate sample size. This is calculated based on the desired confidence level and margin of error.
**Scenario:**
- **Confidence Level:** 95%
- **Margin of Error:** 0.08
The task is to determine how large a sample should be to achieve the desired level of precision without prior data available to set a planning value for \( p^* \).
Upon calculating, a sample size of 150 was entered but marked incorrect.
**Note:**
To find the correct sample size, use the formula for sample size calculation in estimating a population proportion:
\[
n = \left(\frac{Z^2 \cdot p^* \cdot (1-p^*)}{E^2}\right)
\]
Where:
- \( n \) = required sample size
- \( Z \) = Z-score (1.96 for 95% confidence)
- \( p^* \) = estimated proportion (often set to 0.5 if unknown)
- \( E \) = margin of error
Make sure to round up to the nearest whole number after calculation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffae7c33e-d978-40f5-9f98-38011beb158f%2F51725154-4fd0-420a-b65d-e004c20e0209%2F9sgo9g_processed.png&w=3840&q=75)
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