MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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**Constructing a Confidence Interval: A Statistical Approach**

Assume that we want to construct a confidence interval. Do one of the following, as appropriate: (a) find the critical value \( t_{\alpha/2} \), (b) find the critical value \( z_{\alpha/2} \), or (c) state that neither the normal distribution nor the t distribution applies.

Here are summary statistics for randomly selected weights of newborn girls: 

- Sample size (n) = 294
- Mean (\(\bar{x}\)) = 29.4 hg
- Standard deviation (s) = 6.4 hg

The confidence level is 95%.

**Instructions:**
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

**Options:**

**A.** \( t_{\alpha/2} = \)
  
(Round to two decimal places as needed.)

**B.** \( z_{\alpha/2} = \)
  
(Round to two decimal places as needed.)

**C.** Neither the normal distribution nor the t distribution applies.

---

**Explanation of Critical Values:**

- **Critical Value \( t_{\alpha/2} \):** The t-score is used when the sample size is small (typically n < 30) or when the population standard deviation is unknown and the sample standard deviation is used instead. It is derived from the t-distribution.
  
- **Critical Value \( z_{\alpha/2} \):** The z-score, derived from the standard normal distribution, is used when the sample size is large (n > 30) and the population standard deviation is known.
  
- **Choosing Correct Distribution:** Based on the sample size and whether the population standard deviation is known or unknown, one can decide which distribution's critical value to use.
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Transcribed Image Text:**Constructing a Confidence Interval: A Statistical Approach** Assume that we want to construct a confidence interval. Do one of the following, as appropriate: (a) find the critical value \( t_{\alpha/2} \), (b) find the critical value \( z_{\alpha/2} \), or (c) state that neither the normal distribution nor the t distribution applies. Here are summary statistics for randomly selected weights of newborn girls: - Sample size (n) = 294 - Mean (\(\bar{x}\)) = 29.4 hg - Standard deviation (s) = 6.4 hg The confidence level is 95%. **Instructions:** Select the correct choice below and, if necessary, fill in the answer box to complete your choice. **Options:** **A.** \( t_{\alpha/2} = \) (Round to two decimal places as needed.) **B.** \( z_{\alpha/2} = \) (Round to two decimal places as needed.) **C.** Neither the normal distribution nor the t distribution applies. --- **Explanation of Critical Values:** - **Critical Value \( t_{\alpha/2} \):** The t-score is used when the sample size is small (typically n < 30) or when the population standard deviation is unknown and the sample standard deviation is used instead. It is derived from the t-distribution. - **Critical Value \( z_{\alpha/2} \):** The z-score, derived from the standard normal distribution, is used when the sample size is large (n > 30) and the population standard deviation is known. - **Choosing Correct Distribution:** Based on the sample size and whether the population standard deviation is known or unknown, one can decide which distribution's critical value to use.
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