**Study on Skull Sizes Across Time Periods**A study was conducted on the skull sizes of humans during different time periods. The results are shown in the table below. Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Complete parts (a) and (b) below.| Time Period | 4000 B.C. ($ \mu_1 $) | A.D. 150 ($ \mu_2 $) ||-------------|----------------|-----------|| n | 27 | 27 || $ \bar{x} $ | 131.05 mm | 133.94 mm || s | 5.18 mm | 5.27 mm |(a) Use a 0.05 significance level to test the claim that the mean skull breadth in 4000 B.C. is less than the mean skull breadth in A.D. 150.### Hypothesis Testing**Null and Alternative Hypotheses:**- **Option A:** - $ H_0: \mu_1 = \mu_2 $ - $ H_1: \mu_1 > \mu_2 $- **Option B:** - $ H_0: \mu_1 = \mu_2 $ - $ H_1: \mu_1 < \mu_2 $- **Option C:** - $ H_0: \mu_1 = \mu_2 $ - $ H_1: \mu_1 \neq \mu_2 $- **Option D:** - $ H_0: \mu_1 \neq \mu_2 $ - $ H_1: \mu_1 < \mu_2 $**Test Statistic and Conclusion:**- The test statistic, $ t $, is $ \underline{\hspace{3cm}} $. (Round to two decimal places as needed.)- The P-value is $ \underline{\hspace{3cm}} $. (Round to three decimal places as needed.)**Conclusion for the Test:**- **Option A:** Reject the null hypothesis. There is not sufficient evidence to support the claim that the mean skull breadth was less in 4000 B.C. than A.D. 150

4000 B.C. A.D. 150 A study was done on skull sizes of humans during different time periods. The results are shown in the table. Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Complete parts (a) and (b) below. H1 27 H2 27 131.05 mm 133.94 mm 5.18 mm 5.27 mm a. Use a 0.05 significance level to test the claim that the mean skull breadth in 4000 B.C is less than the mean skull breadth in A.D 150. What are the null and alternative hypotheses? O B. Ho: H1 = H2 H4: H1 H2 OC. Ho: H1 = H2 H: H1 # H2 O D. Ho: H1 #H2 H,: H1

4000 B.C. A.D. 150 A study was done on skull sizes of humans during different time periods. The results are shown in the table. Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Complete parts (a) and (b) below. H1 27 H2 27 131.05 mm 133.94 mm 5.18 mm 5.27 mm a. Use a 0.05 significance level to test the claim that the mean skull breadth in 4000 B.C is less than the mean skull breadth in A.D 150. What are the null and alternative hypotheses? O B. Ho: H1 = H2 H4: H1 H2 OC. Ho: H1 = H2 H: H1 # H2 O D. Ho: H1 #H2 H,: H1

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

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Concept explainers

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!

Question

**Study on Skull Sizes Across Time Periods**

A study was conducted on the skull sizes of humans during different time periods. The results are shown in the table below. Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Complete parts (a) and (b) below.

| Time Period | 4000 B.C. ($ \mu_1 $) | A.D. 150 ($ \mu_2 $) |
|-------------|----------------|-----------|
| n | 27 | 27 |
| $ \bar{x} $ | 131.05 mm | 133.94 mm |
| s | 5.18 mm | 5.27 mm |

(a) Use a 0.05 significance level to test the claim that the mean skull breadth in 4000 B.C. is less than the mean skull breadth in A.D. 150.

### Hypothesis Testing

**Null and Alternative Hypotheses:**
- **Option A:**
- $ H_0: \mu_1 = \mu_2 $
- $ H_1: \mu_1 > \mu_2 $

- **Option B:**
- $ H_0: \mu_1 = \mu_2 $
- $ H_1: \mu_1 < \mu_2 $

- **Option C:**
- $ H_0: \mu_1 = \mu_2 $
- $ H_1: \mu_1 \neq \mu_2 $

- **Option D:**
- $ H_0: \mu_1 \neq \mu_2 $
- $ H_1: \mu_1 < \mu_2 $

**Test Statistic and Conclusion:**

- The test statistic, $ t $, is $ \underline{\hspace{3cm}} $. (Round to two decimal places as needed.)
- The P-value is $ \underline{\hspace{3cm}} $. (Round to three decimal places as needed.)

**Conclusion for the Test:**

- **Option A:** Reject the null hypothesis. There is not sufficient evidence to support the claim that the mean skull breadth was less in 4000 B.C. than A.D. 150

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