# Q# 106: Sample size estimation for smart watch users

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**What sample size is required to estimate the proportion of U.S. adults who wear a smart watch with 95% confidence within a 3% error rate? A survey taken one year ago indicated that 15% of all adults wore a smart watch. Note: the z-score for a 95% confidence interval is 1.96.**

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## ANSWER

# Why is Sample Size Important?

Choosing the correct sample size is vital for several reasons:

- Accuracy: A larger sample size reduces the margin of error, leading to more precise estimates.
- Reliability: Adequate sample sizes help ensure that the results are representative of the entire population.
- Cost-Effectiveness: While larger sample sizes provide more accuracy, they also require more resources. Balancing accuracy and resource availability is key.

When conducting surveys to estimate population parameters, one crucial aspect is determining the appropriate sample size. This ensures that the estimate is both accurate and reliable. In this blog post, we’ll explore how to calculate the sample size needed to estimate the proportion of U.S. adults who wear a smart watch, aiming for a 95% confidence level and a 3% margin of error.

# Understanding the Requirements

- Confidence Level: 95%
- Margin of Error (E): 3% (0.03)
- Proportion Estimate (p): 15% (0.15), based on a previous survey
- Z-score for 95% Confidence Level: 1.96

# Formula for Sample Size Calculation

To calculate the sample size for estimating a population proportion, we use the following formula:

𝑛=(𝑍²⋅𝑝⋅(1−𝑝)/𝐸²)

Where:

- 𝑛 is the required sample size.
- 𝑍 is the z-score corresponding to the desired confidence level.
- 𝑝 is the estimated proportion of the population.
- 𝐸 is the margin of error.

# Step-by-Step Calculation

- Identify the z-score: For a 95% confidence level, the z-score is 1.96.
- Use the estimated proportion: The previous survey indicated that 15% of U.S. adults wear a smart watch, so 𝑝=0.15.
- Determine the margin of error: The desired margin of error is 3%, so 𝐸=0.03.
- Plug these values into the formula:

𝑛=(1.962⋅0.15⋅(1−0.15)/0.03²)

- Calculate the components:

- 𝑍2=1.962=3.8416
- 𝑝⋅(1−𝑝)=0.15⋅0.85 = 0.1275
- 𝐸2 = 0.032 = 0.0009

3. Substitute these into the formula:

𝑛=(3.8416⋅0.1275/0.0009)

4. Perform the calculation:

𝑛=(0.489804/0.0009)≈544.22

Since we can’t survey a fraction of a person, we always round up to the next whole number.

𝑛≈545

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