(b) Use the 95% rule and the standard error SE = 0.028 to give an interval which contains roughly 95% of the sample proportions in the sampling distribution.
(b) Use the 95% rule and the standard error SE = 0.028 to give an interval which contains roughly 95% of the sample proportions in the sampling distribution.
MATLAB: An Introduction with Applications
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ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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![### Sampling Distribution of Sample Proportions
#### Overview
The image illustrates a sampling distribution of sample proportions from random samples, each of size \(n = 200\), taken from a population. The dotplot shows how sample proportions are distributed.
#### Dotplot Analysis
- **Title**: Sampling Dotplot of Proportion
- **X-Axis**: Represents sample proportions, ranging from 0.100 to 0.275.
- **Y-Axis**: Counts of sample proportions (frequency), ranging up to 80.
- **Key Metrics**:
- **Number of Samples**: 1000
- **Mean (Average) Proportion**: 0.199
- **Standard Error (SE)**: 0.028
The dotplot displays a bell-shaped distribution centered around the mean proportion of 0.199. This indicates that most sample proportions are clustered around the mean, with fewer samples having extreme values.
#### Tasks
**(a)** *Estimate the Population Proportion*
Use the provided dotplot to estimate \(\hat{p}\), the average value that might indicate the population proportion. Based on the mean indicated (0.199), \(\hat{p}\) is approximately 0.199.
**(b)** *Calculate the 95% Confidence Interval*
Utilize the 95% rule (Empirical Rule) and the provided standard error (SE = 0.028) to determine an interval that encompasses roughly 95% of the sample proportions in the sampling distribution.
- The 95% confidence interval can be calculated as:
\[
\text{Mean} \pm 2 \times \text{SE} = 0.199 \pm 2 \times 0.028
\]
- This results in:
\[
0.199 \pm 0.056
\]
- Thus, the interval is approximately:
\[
[0.143, 0.255]
\]
This interval suggests that we are 95% confident that the true population proportion lies within this range.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F055086f6-4d26-404b-b10d-4e78791428d4%2F356dd0b4-6514-4a14-ba87-b0c4806ecd00%2F8uzn9pi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Sampling Distribution of Sample Proportions
#### Overview
The image illustrates a sampling distribution of sample proportions from random samples, each of size \(n = 200\), taken from a population. The dotplot shows how sample proportions are distributed.
#### Dotplot Analysis
- **Title**: Sampling Dotplot of Proportion
- **X-Axis**: Represents sample proportions, ranging from 0.100 to 0.275.
- **Y-Axis**: Counts of sample proportions (frequency), ranging up to 80.
- **Key Metrics**:
- **Number of Samples**: 1000
- **Mean (Average) Proportion**: 0.199
- **Standard Error (SE)**: 0.028
The dotplot displays a bell-shaped distribution centered around the mean proportion of 0.199. This indicates that most sample proportions are clustered around the mean, with fewer samples having extreme values.
#### Tasks
**(a)** *Estimate the Population Proportion*
Use the provided dotplot to estimate \(\hat{p}\), the average value that might indicate the population proportion. Based on the mean indicated (0.199), \(\hat{p}\) is approximately 0.199.
**(b)** *Calculate the 95% Confidence Interval*
Utilize the 95% rule (Empirical Rule) and the provided standard error (SE = 0.028) to determine an interval that encompasses roughly 95% of the sample proportions in the sampling distribution.
- The 95% confidence interval can be calculated as:
\[
\text{Mean} \pm 2 \times \text{SE} = 0.199 \pm 2 \times 0.028
\]
- This results in:
\[
0.199 \pm 0.056
\]
- Thus, the interval is approximately:
\[
[0.143, 0.255]
\]
This interval suggests that we are 95% confident that the true population proportion lies within this range.
Expert Solution
Introduction
Empirical rule explain the spread of data in the normal distribution. It satisfy the three ranges of data spread within the standard deviation. They are within one standard deviation, within two standard deviation and within three standard deviation. Within one standard deviation explain 68% of data, Within two standard deviation explain 95% of data and Within three standard deviation explain 99.7% of data.
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