**Statistical Hypothesis Testing: Drying Time Analysis**Consider a painting situation in which drying time for test specimens is normally distributed with σ = 7. The hypotheses H₀: μ = 74 and H₁: μ < 74 are to be tested using a random sample of n = 25 observations.(a) **How many standard deviations (of X̅) below the null value is X̅ = 72.3?** - (Round your answer to two decimal places.)(b) **Suppose α = 0.008.**1. **State the conclusion using α = 0.008.**2. **Calculate the test statistic and determine the P-value.** - (Round your test statistic to two decimal places and your P-value to four decimal places.)(c) **For the test procedure with α = 0.006:**1. **What is β(70)?** - (Round your answer to four decimal places.)(d) **If the test procedure with α = 0.006 is used, what n is necessary to ensure that β(70) ≤ 0.010?** - (Round your answer up to the next whole number.)(e) **If a Level 0.01 test is used with n = 100, what is the probability of a Type II error when μ = 75?** - (Round your answer to four decimal places.)**Appendix: Z Tables**You may need to use the appropriate table in the Appendix of Z Tables to answer this question.**Conclusions Explanation:**- **Reject the null hypothesis**: There is sufficient evidence to conclude that the mean drying time is less than 74.- **Do not reject the null hypothesis**: There is insufficient evidence to conclude that the mean drying time is less than 74.**Additional Notes:**- The problem involves calculating and analyzing standard deviations, test statistics, P-values, and probabilities related to Type II errors (β).- Round all calculations appropriately as specified to ensure precision in hypothesis testing.

Answered: Consider a paint-drying situation in…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

**Statistical Hypothesis Testing: Drying Time Analysis**

Consider a painting situation in which drying time for test specimens is normally distributed with σ = 7. The hypotheses H₀: μ = 74 and H₁: μ < 74 are to be tested using a random sample of n = 25 observations.

(a) **How many standard deviations (of X̅) below the null value is X̅ = 72.3?**
- (Round your answer to two decimal places.)

(b) **Suppose α = 0.008.**

1. **State the conclusion using α = 0.008.**
2. **Calculate the test statistic and determine the P-value.**
- (Round your test statistic to two decimal places and your P-value to four decimal places.)

(c) **For the test procedure with α = 0.006:**

1. **What is β(70)?**
- (Round your answer to four decimal places.)

(d) **If the test procedure with α = 0.006 is used, what n is necessary to ensure that β(70) ≤ 0.010?**
- (Round your answer up to the next whole number.)

(e) **If a Level 0.01 test is used with n = 100, what is the probability of a Type II error when μ = 75?**
- (Round your answer to four decimal places.)

**Appendix: Z Tables**

You may need to use the appropriate table in the Appendix of Z Tables to answer this question.

**Conclusions Explanation:**

- **Reject the null hypothesis**: There is sufficient evidence to conclude that the mean drying time is less than 74.
- **Do not reject the null hypothesis**: There is insufficient evidence to conclude that the mean drying time is less than 74.

**Additional Notes:**

- The problem involves calculating and analyzing standard deviations, test statistics, P-values, and probabilities related to Type II errors (β).
- Round all calculations appropriately as specified to ensure precision in hypothesis testing.

Expert Solution

Step 1: Introduction

Note: “Since you have posted a question with multiple sub parts, we will provide the solution only to the first three sub parts as per our Q&A guidelines. Please repost the remaining sub parts separately.”

The formula to calculate z-score is,

$z equals fraction numerator x with bar on top minus mu over denominator open parentheses fraction numerator sigma over denominator square root of n end fraction close parentheses end fraction x with bar on top colon space s a m p l e space m e a n n colon space s a m p l e space s i z e mu colon p o p u l a t i o n space m e a n sigma colon space p o p u l a t i o n space space s tan d a r d space d e v i a t i o n$

The type-II error is nothing but failing to reject the null hypothesis , when it is false.