Suppose the data are independent observations sampled from some population distribution. In which of the following circumstances we would NOT be safe to construct a one-sample t confidence nterval for a population mean based on a simple random sample from a huge population? (Select one answer.) (i) The sample size is 5, and we know from past experience that the population distribution is normal. (ii) The sample size is 10, and the observations are 0, 0, 0, 0, 0, 0, 0 ,1 ,1 ,18. (iii) The sample size is 45 and the boxplot of the data is 0 00 m @ @ m0000 O O 2 6. 10 14 18 (iv) The sample size is 1800 and the histogram of the data is 10 20 30 40 50 (v) It's not safe to construct t confidence intervals in any of the circumstances above.

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**Identifying Safe Circumstances for Constructing a One-Sample t Confidence Interval**

Suppose the data are independent observations sampled from a certain population distribution. In which of the following circumstances would it NOT be safe to construct a one-sample t confidence interval for a population mean based on a simple random sample from a huge population? Select one answer.

1. **The sample size is 5, and we know from past experience that the population distribution is normal.**

2. **The sample size is 10, and the observations are 0, 0, 0, 0, 0, 0, 0, 1, 1, 18.**

3. **The sample size is 45, and the boxplot of the data is:**

    ![Boxplot Image]
    
    Explanation: The boxplot indicates that the data has some mild outliers and possibly some skewness, but provides a visual summary of the distribution.

4. **The sample size is 1800, and the histogram of the data is:**

    ![Histogram Image]
    
    Explanation: The histogram displays the data distribution. It shows a widespread with some skewness and variability in the observations up to a maximum value around 50.

5. **It’s not safe to construct t confidence intervals in any of the circumstances above.**

*Note:* Based on statistical guidelines, the key is to understand under what conditions the assumptions of the t-distribution (normality with small sample sizes, robustness to non-normality with larger sample sizes) hold true. 

**Detailed Figures Explanation:**

- **Boxplot (iii)**: A boxplot visual representation shows quartiles and potential outliers. It gives a quick view of the data's central tendency, spread, and symmetry.
    - The central line within the box signifies the median.
    - The edges of the box indicate the interquartile range (IQR).
    - Whiskers extend to the smallest and largest values within 1.5*IQR from the quartiles. Points beyond these are considered outliers.
  
- **Histogram (iv)**: A histogram is a graphical representation of data distribution.
    - The x-axis indicates the value bins, while the y-axis shows frequency.
    - The spread and height of the bars illustrate variability and distribution of the data.
  
Choosing the right scenario often requires balancing between normalcy assumptions and the central limit theorem, particularly related
Transcribed Image Text:**Identifying Safe Circumstances for Constructing a One-Sample t Confidence Interval** Suppose the data are independent observations sampled from a certain population distribution. In which of the following circumstances would it NOT be safe to construct a one-sample t confidence interval for a population mean based on a simple random sample from a huge population? Select one answer. 1. **The sample size is 5, and we know from past experience that the population distribution is normal.** 2. **The sample size is 10, and the observations are 0, 0, 0, 0, 0, 0, 0, 1, 1, 18.** 3. **The sample size is 45, and the boxplot of the data is:** ![Boxplot Image] Explanation: The boxplot indicates that the data has some mild outliers and possibly some skewness, but provides a visual summary of the distribution. 4. **The sample size is 1800, and the histogram of the data is:** ![Histogram Image] Explanation: The histogram displays the data distribution. It shows a widespread with some skewness and variability in the observations up to a maximum value around 50. 5. **It’s not safe to construct t confidence intervals in any of the circumstances above.** *Note:* Based on statistical guidelines, the key is to understand under what conditions the assumptions of the t-distribution (normality with small sample sizes, robustness to non-normality with larger sample sizes) hold true. **Detailed Figures Explanation:** - **Boxplot (iii)**: A boxplot visual representation shows quartiles and potential outliers. It gives a quick view of the data's central tendency, spread, and symmetry. - The central line within the box signifies the median. - The edges of the box indicate the interquartile range (IQR). - Whiskers extend to the smallest and largest values within 1.5*IQR from the quartiles. Points beyond these are considered outliers. - **Histogram (iv)**: A histogram is a graphical representation of data distribution. - The x-axis indicates the value bins, while the y-axis shows frequency. - The spread and height of the bars illustrate variability and distribution of the data. Choosing the right scenario often requires balancing between normalcy assumptions and the central limit theorem, particularly related
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