Show that if Y_1, ...., Y_n be independent random variables with moment-generati

Question

Show that if Y_1, ...., Y_n be independent random variables with moment-generating functions m_y_1 (t), ..., m_y_2 (t), respectively. If U = Y_1 + Y_2 + .....+ Y_n, then m_U(t) = m_y_1 (t) m_y_2(t) ....m_y_3(t) = Product^_i = 1 m_y_2(t). The output voltage for a certain electric circuit is specified to be 130. A sample of 40 independent readings on the voltage for this circuit gave a sample mean of 128.6 and a standard deviation of 2.1. Test the a test hypothesis that the average output voltage is 130 against the alternative that it is less than 130. Use with level 0.05.

Explanation / Answer

Solution:

Here, we have to use one sample t test for population mean.

H0: µ = 130

Ha: µ < 130

This is a one tailed test. This is lower tailed or left tailed test.

We are given

Level of significance = alpha = 0.05

Sample size = n = 40

Sample mean = Xbar = 128.6

Sample standard deviation = S = 2.1

Degrees of freedom = n - 1 = 40 - 1 = 39

The test statistic formula is given as below:

t = (Xbar - µ) / [S/sqrt(n)]

t = (128.6 – 130)/[2.1/sqrt(40)] = -4.2164

Lower critical value = -1.6849

P-value = 0.0001

Alpha value = 0.05

P-value < Alpha value

So, we reject the null hypothesis

There is sufficient evidence to conclude that the average voltage for a certain electric circuit is less than 130.

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