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So for this, we will be doing a system of equations, with one equation representing the total invested and the other equation representing the total gained from the investments. But first, we have to do calculations:
firstly, to represent a percentage gain (in this case 12% gain) you are to add 0.12 (12% in decimal form) to 1 to get 1.12. <em>Keep 1.12 in mind, as it will be used for the equations.</em>
Next, to represent a percentage loss (in this case 11%), you are to subtract 0.11 (11% in decimal form) from 1 to get 0.89. <em>Keep 0.89 in mind, as it will be used in the equations.</em>
Next, we need to find the total amount of money after the investments. To do this, add 21,000 and 1,715 together to get 22,715. <u>$22,715 is the total amount of money gained in 1 year.</u>
Now that we have all of our information, we can form our equations as such:
- Let x = $ invested into the account with 12% gain and y = $ invested into the account with 11% loss
Now with these system of equations, I will be using the substitution method. So firstly, with the first equation subtract x on both sides of that equation:
Now that we know that y = 21000 - x, replace y for (21000 - x) in the second equation and solve for x as such:
Now that we have the value of x, substitute it into either equation to solve for y:
<u>In short, $17500 was invested into the account that gained 12% and $3500 was invested into the account that had lossed 11%.</u>
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Answer:
Since our calculated value is outside than our critical value,, we have enough evidence to reject the null hypothesis at 5% of significance. So there is not nough evidence to conclude that mean for Ohio it's significantly different than the mean for Georgia.
Step-by-step explanation:
1) Data given and notation
represent the mean for Georgia
represent the mean for Ohio
represent the population standard deviation for Georgia
represent the population standard deviation for Ohio
sample size for Georgia
sample size for Ohio
z would represent the statistic (variable of interest)
represent the p value
2) Concepts and formulas to use
We need to conduct a hypothesis in order to check if the means are different for both groups, the system of hypothesis would be:
H0:
H1:
If we analyze the size for the samples both are higher than 30, we can assume that the normal distribution is the correct disirbution for the statistic, so for this case is better apply a z test to compare means, and the statistic is given by:
(1)
z-test: Is used to compare group means. Is one of the most common tests and is used to determine whether the means of two groups are equal to each other.
3) Calculate the statistic
We have all in order to replace in formula (1) like this:
4) Find the critical value
In order to find the critical value we need to take in count that we are conducting a two tailed test, so we are looking on the normal standard distribution a value that accumulates 0.025 of the area on the right and 0.975 of the area on the left. We can us excel or a table to find it, for example the code in Excel is:
"=NORM.INV(1-0.025,0,1)", and we got
5) Statistical decision
Since our calculated value is outside than our critical value,, and outside of (-1.96,1.96), we have enough evidence to reject the null hypothesis at 5% of significance. So there is not nough evidence to conclude that mean for Ohio it's significantly different than the mean for Georgia.