Simplify to it's lowest form. 5 * 6 / (1 +5) * (9/3)
It's important to remember our order of operations, Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction. We have to do our parenthesis first. This simplifies to 5 * 6/ (6) * (3) Now we need to do our multiplication and subtraction. This simplifies to 30 / 18 Now we have an improper fraction on our hands. We have two options, we could convert to decimal for or we could simplify the fraction. In this example, let's simplify the fraction, since the instructions said "Simply". Both 30 and 18 are divisible by 2, so let's start there. 30 / 2 = 15 and 18 / 2 = 9 So now our fraction is 15 / 9 Both 15 and 9 are divisible by 3, so let's simplify some more. 15 / 3 = 5 9 / 3 = 3 So now our fraction is 5 / 3 And that's as simple as it gets!
We have two samples of data taken from the height of sunflowers growing in two different kinds of soils. One type of soil is the normal soil, or the control, and the other is grown in fertilized soil. We want to find if there is any significant difference in the height of the two sunflowers. The data set for the normal soil flowers is: 1.1 1.3 2 3 1.2 3.2 The data set for the fertilized group of flowers is 2.5 1.3 .7 2 3.3 5.2
The way we see if there's any significance between the two data sets is to do a hypothesis test. For our null hypothesis we assume there is no difference, so we say the means have no difference. For our alternative hypothesis we assume there is a difference, so the means are different. Now that we have established our hypotheses, we need to test it. With the small sample sizes we have, it is best to do a Student's T-test. We could also perform an F test, but for this example we will use Student's T. We are also going to assume we are calculating at 95% confidence The variances of the two samples are assumed to be the same, so we are going to pool the variances. First we must calculate the variances of the two samples. This could be done by hand, but for the sake of the example, we are going to do it in Excel. Using the formula =VAR.S and highlighting our first data group, we see that that variance of that group is .875. Repeating the process, we find the variance of the second group is 2.572. The steps for calculating the pooled variance are to multiply the variances by their respective degrees of freedom, adding them, and dividing by the sum of the data points minus the number of groups. This looks like (.875 * 5) + (2.572 * 5) / (12 - 2). The pooled variance works out to 1.723. Now that we have our pooled average, we can calculate our t-score. To do that, we take the average of the first group, subtract it by the average of the second group, and divide it by the standard error, which in this case is he pooled variance times the square root of 1 divided the sample size of the first group plus 1 divided the sample size of the second group. The average of the first group is 1.97 and the average of the second group is 2.5. This calculation looks like: 1.97 - 2.5 / 1.723 * sqrt(1/6 + 1/6) OR 1.97 - 2.5 / sqrt(1.723/6 + 1.723/6) Our t-score works out to 1.329. Now we have to find our critical value. We could do that by looking at a table, but for the sake of our example, let's use Excel again. With the formula =T.INV.2T(.05,10), we see the critical value for an alpha of .05 and 10 degrees of freedom, the critical value is plus or minus 2.228. We could also find the p-value of our t-score and use that to make a decision about our hypotheses Since our t-score is less than our critical value, we do not reject the null hypothesis. We can not conclusively say that the means of the two groups are any different, or in real world terms, that the fertilizer works .
Suppose we are modeling the market for t-shirts using the basic supply and demand framework. The government suddenly decides to remove all cotton subsidies, causing the price of cotton to skyrocket. What is the effect on the aggregate demand and aggregate supply of t shirts? What's the effect on the equilibrium price and the quantity supplied?
The supply of t-shirts will fall due to to the fact that cotton is a input of t-shirts, and an increase in the price of an input will cause the output's supply to fall. Demand will not be effected. Price will increase, and quantity supplied will fall.