An automatic character recognition device successfully reads about 8383​% of handwritten credit card applications. To estimate what might happen when this device reads a stack of​ applications, the company did a simulation using samples of size​ 20, 50,​ 75, and 100. For each sample​ size, they simulated 1000 samples with success rate pequals=0.830.83 and constructed the histogram of the 1000 sample proportions. Explain what these histograms say about the sampling distribution model for sample proportions.

The histograms generated from the simulation of 1000 samples for each sample size (20, 50, 75, and 100) provide insight into the **sampling distribution model for sample proportions**. Let's break down what these histograms are showing:

### 1. **Shape of the Distribution**:
   - As the sample size increases (from 20 to 100), the shape of the histogram becomes more **bell-shaped** and **symmetrical**, resembling a normal distribution. This is consistent with the Central Limit Theorem, which states that the sampling distribution of the sample proportion will approximate a normal distribution as the sample size becomes large, provided that the underlying population proportion is not too close to 0 or 1.
   
### 2. **Spread of the Distribution**:
   - The spread (or standard deviation) of the sample proportions decreases as the sample size increases. This means that larger samples yield more precise estimates of the true proportion (in this case, 83%). The **spread decreases** because the standard error of the sample proportion is given by:
     \[
     SE = \sqrt{\frac{p(1-p)}{n}}
     \]
     where \( p = 0.83 \) and \( n \) is the sample size. As \( n \) increases, the standard error decreases, leading to less variability in the sample proportions.

### 3. **Center of the Distribution**:
   - In all histograms, regardless of the sample size, the **center** (mean) of the distribution is approximately 0.83, which is the true population proportion. This shows that the sample proportion is an **unbiased estimator** of the population proportion, meaning on average, the sample proportion will correctly estimate the population proportion.

### 4. **Effect of Sample Size**:
   - For **small sample sizes** (like 20), the histogram is more spread out, meaning there is more variability in the sample proportions. As the sample size increases (like 50, 75, and 100), the histograms become narrower, indicating less variability. This shows that larger sample sizes provide more **consistent** and **reliable** estimates of the population proportion.

### Conclusion:
The histograms reflect that as sample size increases, the sampling distribution of the sample proportions becomes more normal, with less variability, while maintaining the same central value (0.83). This behavior is predicted by the theory of sampling distributions, and it highlights the importance of using larger sample sizes to achieve more accurate estimates.

### Do you want more details or have any questions?

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Here are 5 related questions to consider:
1. How does the Central Limit Theorem apply to this scenario?
2. How can we calculate the standard error for different sample sizes?
3. What are the conditions required for the sampling distribution of proportions to be approximately normal?
4. How does increasing the sample size affect the margin of error in estimating proportions?
5. How would the sampling distribution change if the population proportion were closer to 0 or 1?

**Tip**: Always check that the sample size is large enough to meet the normality condition for proportions: \( np \geq 10 \) and \( n(1-p) \geq 10 \). This ensures that the sampling distribution is approximately normal.

An automatic character recognition device successfully reads about 83% of handwritten credit card applications. To estimate what might happen when this device reads a stack of applications, the company did a simulation using samples of size 20, 50, 75, and 100. For each sample size, they simulated 1000 samples with success rate p=0.83 and constructed the histogram of the 1000 sample proportions. Explain what these histograms say about the sampling distribution model for sample proportions.

Explore the sampling distribution of sample proportions for an automatic character recognition device that successfully reads 83% of credit card applications. This detailed solution breaks down key concepts such as the Central Limit Theorem and standard error for sample sizes 20, 50, 75, and 100. Ideal for college and university students studying statistics.

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