Question 1137293
<font face="times" color="black" size="3">
I'll do the first part to get you started.



Population mean = mu = 40
Population standard deviation = sigma = 10
Because we know the value of the population standard deviation, we can use a normal Z distribution. 



The formula for the z test statistic is 
z = (M-mu)/(sigma/sqrt(n))
where n is the sample size and M is the sample mean (in this case M = 46).



Plug in the values mentioned and simplify
z = (M-mu)/(sigma/sqrt(n))
z = (46-40)/(10/sqrt(n))
z = 6/(10/sqrt(n))
z = (6/1)/(10/sqrt(n))
z = (6/1)*(sqrt(n)/10)
z = (6*sqrt(n))/10
z = (6/10)*sqrt(n)
z = (3/5)*sqrt(n)


Use a table or a calculator to find that z = 1.96 is the approximate critical value that corresponds to a 95% confidence interval, or put another way, you have 5% of the area in the two tails combined (2.5% in each tail). You can use <a href ="http://onlinestatbook.com/2/calculators/inverse_normal_dist.html">this calculator</a> if you don't have a TI calculator. 


If you end up using the calculator I posted in the link, then follow these steps
<ol>
<li>Type 0.05 for the area</li>
<li>Enter 0 for the mean</li>
<li>Enter 1 for the standard deviation (SD)</li>
<li>click on the "outside" radio button</li>
<li>click the "recalculate" button</li>
</ol>
The result "-1.96 and 1.96 will show up where you clicked on the "outside" ratio button. This means that P(Z < -1.96 or Z > 1.96) = 0.05 which is the combined area in the two tails. This combined area in the two tails is the alpha significance level we want.


We'll only focus on the positive z value. This is because M = 46 is larger than mu = 40, so the difference M-mu is positive, meaning the z score is above 0. Any z score larger than 1.96 will be considered statistically significant. 



Plug z = 1.96 into the equation we found earlier. Isolate n
z = (3/5)*sqrt(n)
1.96 = (3/5)*sqrt(n)
1.96*5 = 3*sqrt(n) ..... multiply both sides by 5
9.8 = 3*sqrt(n)
3*sqrt(n) = 9.8
sqrt(n) = 9.8/3 ....... divide both sides by 3
sqrt(n) = 3.2667
n = (3.2667)^2 ...... square both sides
n = 10.6713
Keep in mind that the sample size n can only be a whole number. 



Let's see what happens when n = 10
z = (3/5)*sqrt(n)
z = (3/5)*sqrt(10) ..... replace n with 10
z = 1.89736659610102
this value is <u>not</u> larger than z = 1.96, so a sample size of n = 10 tells us that a sample mean of M = 46 is <u>not</u> statistically significant.



Now plug in n = 11
z = (3/5)*sqrt(n)
z = (3/5)*sqrt(11)
z = 1.98997487421323
this value is larger than z = 1.96



So any sample size of n = 11 or larger will produce a z score that is larger than z = 1.96, meaning that we have a statistically significant result. 



This is just for the sample mean M = 46. The steps for M = 43 will be similar.</font>