Question

For the hypothesis test H0: μ = 10 against H1: μ >10 and variance known, calculate the Pvalue for each of the following test statistics.(a) z0 = 2.05 (b) z0 = −1.84 (c) z0 = 0.4

Answer 1

Answer:

a) $p_v =P(Z>2.05)=1-P(z<2.05)=1-0.980=0.0202$

b) $p_v =P(Z>-1.84)=1-P(z<-1.84)=1-0.0330=0.967$

c) $p_v =P(Z>0.4)=1-P(z<0.4)=1-0.655=0.345$

Step-by-step explanation:

Some previous concepts

The p-value is the probability of obtaining the observed results of a test, assuming that the null hypothesis is correct.

A z-test for one mean "is a hypothesis test that attempts to make a claim about the population mean(μ)".

The null hypothesis attempts "to show that no variation exists between variables or that a single variable is no different than its mean"

The alternative hypothesis "is the hypothesis used in hypothesis testing that is contrary to the null hypothesis"

Hypothesis

Null hypothesis: $\mu=10$

Alternative hypothesis: $\mu >10$

If the random variable is distributed like this: $X \sim N(\mu,\sigma)$

We assume that the variance is known so the correct test to apply here is the z test to compare means, the statistic is given by the following formula:

$z_o=\frac{\bar X -\mu}{\sigma}$

Since we have the values for the statistic already calculated we can calculate the p value using the following formulas:

Part a

$p_v =P(Z>2.05)=1-P(z<2.05)=1-0.980=0.0202$

And in order to find the answer using excel we can use the following code:

"=1-NORM.DIST(2.05,0,1,TRUE)"

Part b

$p_v =P(Z>-1.84)=1-P(z<-1.84)=1-0.0330=0.967$

And in order to find the answer using excel we can use the following code:

"=1-NORM.DIST(-1.84,0,1,TRUE)"

Part c

$p_v =P(Z>0.4)=1-P(z<0.4)=1-0.655=0.345$

And in order to find the answer using excel we can use the following code:

"=1-NORM.DIST(0.4,0,1,TRUE)"

Conclusions

If we use a reference value for the significance, let's say $\alpha=0.05$. For part a the $p_v<\alpha$ so then we can reject the null hypothesis at this significance level.

For part b the $p_v>\alpha$ so then we FAIL to reject the null hypothesis at this significance level.

For part c the $p_v>\alpha$ so again we FAIL to reject the null hypothesis at this significance level.