Question

Part II: Free Response — 1 question. This section is take-home and accounts for one-third of the overall grade on this exam. Please write or type your responses legibly on separate paper. Turn in a hard copy including (1) this page as a cover sheet and (2) your responses. You must show your work to support your solutions; answers provided without necessary support will not receive credit. Collaboration is expressly forbidden. Unless other arrangements are made and approved by the instructor, your responses are due, without exception, at the beginning of class on November 26, 2018.
An electrical firm produces surge protectors for use in industrial applications. These surge protectors are designed to “trip” (activate) when the voltage detected exceeds 120 volts for a specified amount of time. However, the firm suspects that one of the machines it uses to manufacture these surge protectors has not been producing them consistently after a system upgrade. The firm wishes to determine if the surge protectors made on this particular machine have a mean trip voltage that is significantly different from 120 volts: if the trip voltage is too low, it will trip prematurely; if the trip voltage is too high, the system it is designed to protect could be damaged.
Assume that the trip voltages are independent and normally distributed. However, the variance in the lifetimes is not assumed to be equal to that of the population and is thus treated as unknown. Note: this data set is unique to you, and the use of a data set meant for another student will result in a score of 0 for the entirety of this Free Response portion. If your data set is not available on Blackboard, please contact the instructor immediately.
(a) State the null and alternative hypotheses.
(b) Compute the sample mean and sample standard deviation for the data. You may use do so by hand (i.e., by using a calculator) or using technology; the Excel functions AVERAGE and STDEV.S may be helpful.
(c) Indicate the distribution (including the degrees of freedom, if necessary) of the test statistic, and compute its observed value.
(d) Determine the rejection region by finding the critical value(s). Use the 5% significance level.
(e) Based on your test statistic and rejection region, state your conclusion in the context of the question of interest.
(f) Construct a 95% confidence interval for µ. (As a hint, you already determined the critical value you need for this calculation earlier.) Does the interval contain the value 120? Briefly explain how this confidence interval can lead you to the same conclusion as the hypothesis test.

Voltage

123.67

119.45

126.19

123.41

119.39

116.87

125.21

119.82

Answer 1

a) Null hypothesis against the alternative

b)

c) The distribution of test statistic is t distribution. As the population standard deviation is unknown, We apply the t test for testing the above hypothesis For this sample of 8 the degress of freedom is 7

d) The test statistic is

The critical value calculated for 7 degrees of freedom and 5% significance level from the t table is 2.365> 1.498, so the null hypothesis cannot be rejected at this level of significance

e)The region of rejection is the compliment of the interval (-2.365,2.365)

f) 95% confidence interval for the population mean height is

(118.98,124.51). Yes the interval contains 120