**Applied Statistics & Probability for Engineering Technology**

**TAP 5**

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Date: / /2020 |

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*Note: For all the questions, provide detailed solution steps.*

**Exercise 1 (15 points, 5 each)**

A textile fiber manufacturer is investigating a new drapery yarn, which the company claims has a mean thread elongation of 15 kg with standard deviation of 2 kg. The company wishes to test the hypothesis against , using a random sample of four specimens.

- What is the type I error probability if the critical region is defined as kg?
- Find the β for the case in which the true mean is 14.25 kg.
- Find the power for the case in which the true mean is 14.11.

**Exercise 2 (20 points, 5 each)**

One – sample Z:

Variable | N | Mean | SE Mean | Standard Deviation | Variance | P-Value | |

x | 20 | 50.184 | ? | 1.815 | ? | ? | ? |

Test of μ = 12 vs μ > 12. The assumes σ = 0.77.

- Find the missing values.
- Is this a one – sided or a two – sided test?
- Use the normal table and the preceding data to construct a 95% two – sided CI of the mean.
- What would the P – value be if the alternative hypothesis is μ ≠ 17.

**Exercise 3 (15 points, 5 each)**

A melting point test samples of a binder used in a manufacturing rocket propellant resulted in . Assume that the melting point is normally distributed with . We want to perform a hypothesis test, with a 0.05 level of significance, to determine whether the service goal of is being achieved.

- Write the hypothesis test that you have to examine and determine if it is a one – sided or a two – sided test?
- Find the P-value and the critical value?
- Define if we have to reject or accept the null hypothesis test with two ways.

**Exercise 4 (50 points, 5 each)**

A bank wants to check the saving behaviour of the clients. We consider the yearly saving behaviour of the clients is depending from the annual salary (unit is salary in thousands of euros). We have a random sample of ten clients in the below table:

(annual salary in thousands) | 50 | 31 | 28 | 45 | 50 | 32 | 36 | 55 | 26 | 47 |

(annual savings in thousands) | 5 | 3 | 6 | 4 | 6 | 1 | 2 | 8 | 2 | 3 |

Answer the following questions:

- Make the scatter plot of X and Y, graph the trendline.
- Find the covariance.
- Find the correlation coefficient.
- Interpret the correlation coefficient.
- Find the estimated linear regression model.
- Interpret the estimated coefficient
- Interpret the estimated coefficient
- Find the estimated valued of the number of savings, if someone has annual salary of 30 thousand euros.
- Find the coefficient of determination.
- Interpret the coefficient of determination.

**Bonus Exercise 5 (10 points, 5 each)**

The sodium content of ten 250 – gram boxes of organic cornflakes was determined. The level of significance is 0.05. The data (in milligrams), are as follows:

129.74, 128.77, 133.15, 129.73, 130.8, 130.92, 129.78, 130.12, 129.39, 129.00

- Write the hypothesis test if we want to claim that the mean sodium content of this brand of cornflakes is less than 130.5.
- Is this a one – sided or a two – sided test?
- Find the critical value?
- Define if we have to reject or accept the null hypothesis test.

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