91Ó°ÊÓ

Find the sample size required to estimate the population mean. Data Set 1 "Body Data" in Appendix B includes weights of 153 randomly selected adult males, and those weights have a standard deviation of \(17.65 \mathrm{kg}\). Because it is reasonable to assume that weights of male statistics students have less variation than weights of the population of adult males, let \(\sigma=17.65 \mathrm{kg}\). How many male statistics students must be weighed in order to estimate the mean weight of all male statistics students? Assume that we want \(90 \%\) confidence that the sample mean is within \(1.5 \mathrm{kg}\) of the population mean. Does it seem reasonable to assume that weights of male statistics students have less variation than weights of the population of adult males?

Step by step solution

01

Identify The Variables

Extract the given values: - Population standard deviation \( \sigma = 17.65 \) - Margin of error (E) = 1.5 kg - Confidence level = 90%.

02

Determine the Z-score for the Confidence Level

For a 90% confidence level, the corresponding Z-score is 1.645. This value is obtained from the Z-table or standard normal distribution table.

03

Use the Sample Size Formula

The formula to find the required sample size (n) is: \[ n = \left(\frac{Z \cdot \sigma}{E}\right)^2\] Plug in the values: \[ n = \left(\frac{1.645 \cdot 17.65}{1.5}\right)^2 \]

04

Calculate the Sample Size

Perform the calculation: \[ n = \left(\frac{1.645 \cdot 17.65}{1.5}\right)^2 \approx \left(19.48\right)^2 \approx 379.4 \] Since the sample size must be a whole number, round 379.4 up to 380.

05

Draw Conclusion About The Assumption

We assume that the weights of male statistics students have less variation than weights of the population of adult males. The given problem states that it's reasonable to assume this.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Mean Estimation

Population mean estimation is about figuring out the average value (mean) of a particular attribute in a population. This mean is the central value that indicates the general tendency of the data. However, we usually can't measure the entire population, so we estimate the mean from a sample. The formula used is: \[\bar{x} = \frac{\text{Total sum of all sample values}}{n}\] where \(\bar{x}\) is the sample mean and \(n\) is the sample size. Usually, knowing the mean helps in understanding the overall trend of the population data.

Confidence Level

When estimating population parameters, we often talk about a confidence level. This indicates the degree of certainty we have that our sample statistic falls within a specific range of the true population parameter. For instance, a 90% confidence level implies that we are 90% certain that the true population mean lies within our specified range. Generally:

• Common confidence levels are 90%, 95%, and 99%.
• The higher the confidence level, the wider the range.

In practice, you’ll see this used to set confidence intervals around the estimate, often represented by Z-scores in a standard normal distribution.

Margin of Error

Margin of error tells you how much you can expect your sample results to differ from the actual population mean. It is the ‘plus-minus’ figure usually reported in survey results. Mathematically, it is captured in formulas as ‘E’. The margin of error (E) gets smaller with larger sample sizes, indicating more precise estimates. For instance:

• If our margin of error is 1.5 kg, our estimates will be within 1.5 kg of the population mean 90% of the time.
• The formula incorporating margin of error in sample size estimation is \(n = \frac{Z^2 \times \text{variation}}{E^2}\).

This allows you to balance between sample size and the desired precision of your estimate.

Normal Distribution

Normal distribution is a bell-shaped curve that represents the spread of a dataset. Most values cluster around the mean and the probability of values decreases as they move away. This makes the mean, median, and mode all fall at the center of the distribution. When estimating population parameters:

• The data should ideally follow a normal distribution.
• It helps in using Z-scores for determining probabilities and confidence levels.

For instance, 90% confidence level corresponds to a Z-score of 1.645 in a standard normal distribution. This concept is crucial as many statistical methods rely on normal distribution for accurate estimation and decision making.