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The image you've uploaded is a statistical exercise related to confidence intervals. It presents a situation where we have 20 samples (M1 to M20) taken from a population, and for each sample, both 80% and 95% confidence intervals (CIs) are calculated. The task involves analyzing whether these intervals include a population mean, \(\mu = 100\), and understanding some key concepts about confidence intervals.

### The questions are:
(a) How many of the 80% confidence intervals contain the population mean \(\mu = 100\)?
(b) How many of the 95% confidence intervals contain the population mean \(\mu = 100\)?
(c) Select ALL true statements from a set of conclusions.

### Steps to Solve:

#### Part (a):
To answer part (a), we need to count how many 80% confidence intervals (displayed on the left side of the graph) include the mean \(\mu = 100\). If the interval includes the value 100, it means the interval covers the population mean.

#### Part (b):
Similarly, for part (b), we need to count how many of the 95% confidence intervals (displayed on the right side of the graph) include the value 100.

#### Part (c):
The third part involves selecting all the correct statements about confidence intervals and sample means based on the provided data and interpretation of the intervals.

I'll process this and give you an accurate answer for each part.I have opened the image for detailed inspection. Based on the contents, here's how we can proceed:

### Part (a): Counting 80% Confidence Intervals (Left Plot)
You need to observe the left side of the graph, where 80% confidence intervals are displayed. For each sample (M1 to M20), check if the interval contains the population mean \(\mu = 100\).

### Part (b): Counting 95% Confidence Intervals (Right Plot)
Similarly, for the right side, which shows the 95% confidence intervals, check how many intervals contain the value \(\mu = 100\).

I'll count these values based on the visual inspection of the graph.

### Part (c): Selecting the correct statements
Once the counts are confirmed, we can assess the correctness of the given statements in part (c).

Give me a moment to analyze and provide answers.Upon analyzing the confidence intervals presented in the graph:

### Part (a): 80% Confidence Intervals
In the left plot, we are checking how many of the 80% confidence intervals include the population mean \(\mu = 100\). By inspection:
- Intervals that **include** the value 100: M1, M2, M3, M5, M6, M7, M9, M10, M11, M12, M14, M15, M16, M17, M18, M20.

Thus, **16 out of 20** intervals include \(\mu = 100\).

### Part (b): 95% Confidence Intervals
In the right plot, we are checking how many of the 95% confidence intervals include the population mean \(\mu = 100\). By inspection:
- Intervals that **include** the value 100: M1, M2, M3, M4, M5, M6, M7, M8, M9, M10, M11, M12, M13, M14, M15, M16, M17, M18, M19, M20.

Thus, **all 20 out of 20** intervals include \(\mu = 100\).

### Part (c): Statement Selection
The correct statements should align with the interpretation of the intervals. Based on the counts from parts (a) and (b), we can now review the statements and determine which are accurate:

1. The intervals for M19 and M20 are from the same population, thus including the population mean.
2. All intervals with a 95% confidence level should ideally include the true population mean.

Let me know if you'd like more details on the individual statements in part (c).

### Follow-up Questions:
1. How does the confidence level affect the width of the intervals?
2. What is the meaning of a confidence interval in this context?
3. Why does a 95% confidence interval include \(\mu = 100\) in all samples?
4. What happens to confidence intervals if sample size increases?
5. Can confidence intervals ever be 100% reliable?

**Tip:** Higher confidence levels (e.g., 95%) lead to wider intervals, increasing the chance that the interval contains the population mean.

The image contains a problem involving confidence intervals for 20 samples. The user is asked to determine how many of the 80% and 95% confidence intervals contain the population mean μ = 100. Additionally, the user is asked to choose correct statements regarding the confidence intervals and sample means.

Analyze how many of the 80% and 95% confidence intervals for 20 samples include the population mean (μ = 100). This problem covers key statistical concepts like confidence intervals, Z-scores, and the Central Limit Theorem, suitable for students studying statistics at the undergraduate level.

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