Question

directions: beginning in the cell marked #1, you will complete the problem and then find the answer. this will lead you to the next problem which will be problem #2. complete the circuit in the following manner until you end up at the 1st problem and complete the circuit.
ans: 2.89
#1
the scores on an ap stats tests are normally distributed with a mean of 84 and a standard deviation 3. use the empirical rule to determine the percent of students that scored higher than a 87.
ans: 14.1%
many employees were surveyed at a large corporation. they were found to have an average salary of 75,000$ with a standard deviation of 5,250$. what is the percentage of employees that make between 72,500$ and 76,000$
ans: 292.3
a chip company wants to ensure the utmost accuracy when filling bags (12oz) with chips. the bags can be overfilled only 2% of the time. the quantity of chips is normally distributed with a standard deviation of .04. what would be the recommended average amount of chips to put in each bag?
ans: .02
a soda shop claims that the distribution of sugar in their sodas is normally distributed with a mean of 9 grams. a randomly selected soda was picked and was determined to have a sugar content of 12.4 grams. this was higher than 88% of sodas. what is the standard deviation of this distribution?
ans: 81.5%
a particular bird in the amazon rainforest has an average bone length of 14.2mm. they have a standard deviation of 1.3 mm. what percent of birds have a bone length of less than 13.1mm?
ans: 17.7
the weights of grapefruits are normally distributed with a mean of 9.64 oz and have a standard deviation of 1.26 oz. what is the 10th percentile of the weights of all grapefruits?

Explanation:

Problem 1:

Step1: Calculate the z - score

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $\mu = 84$ (mean), $\sigma = 3$ (standard deviation) and $x = 87$. So, $z=\frac{87 - 84}{3}=\frac{3}{3}=1$.

Step2: Use the empirical rule

The empirical rule for a normal distribution states that about 68% of the data lies within $z=\pm1$ of the mean. So the percentage of data above $z = 1$ is $\frac{100 - 68}{2}=16\%$.

Problem 2:

Step1: Calculate z - scores

For $x_1=72500$, $z_1=\frac{72500 - 75000}{5250}=\frac{- 2500}{5250}\approx - 0.48$. For $x_2 = 76000$, $z_2=\frac{76000 - 75000}{5250}=\frac{1000}{5250}\approx0.19$.
Using a z - table or calculator, the area to the left of $z_1\approx0.3156$ and the area to the left of $z_2\approx0.5753$. The percentage between them is $0.5753-0.3156 = 0.2597\approx26\%$. But the given answer is $14.1\%$, there might be some approximation differences in using empirical - rule - like shortcuts or z - table look - ups.

Problem 3:

Step1: Find the z - score corresponding to the upper 2%

The upper 2% means an area of 0.98 to the left. Looking up in the z - table, the z - score $z\approx2.05$.

Step2: Use the z - score formula to find the mean

We know $z=\frac{x-\mu}{\sigma}$, where $x = 12$, $\sigma=0.04$ and $z = 2.05$. Rearranging for $\mu$, we get $\mu=x - z\sigma=12-2.05\times0.04=12 - 0.082 = 11.918\approx11.92$. But the given answer is $292.3$ which seems incorrect based on the correct approach.

Problem 4:

Step1: Find the z - score corresponding to 88%

The area to the left is 0.88. Looking up in the z - table, $z\approx1.175$.

Step2: Use the z - score formula to find the standard deviation

We know $z=\frac{x-\mu}{\sigma}$, where $x = 12.4$, $\mu = 9$ and $z = 1.175$. Rearranging for $\sigma$, we get $\sigma=\frac{x-\mu}{z}=\frac{12.4 - 9}{1.175}=\frac{3.4}{1.175}\approx2.9$. But the given answer is $0.02$ which seems incorrect.

Problem 5:

Step1: Calculate the z - score

$z=\frac{13.1 - 14.2}{1.3}=\frac{-1.1}{1.3}\approx - 0.85$.

Step2: Find the percentage

Using a z - table, the area to the left of $z=-0.85$ is approximately $0.1977\approx20\%$. But the given answer is $81.5\%$ which is incorrect.

Problem 6:

Step1: Find the z - score for the 10th percentile

Looking up in the z - table, the z - score corresponding to the 10th percentile is approximately $z=-1.28$.

Step2: Use the z - score formula to find the value

We know $z=\frac{x-\mu}{\sigma}$, where $\mu = 9.64$, $\sigma = 1.26$ and $z=-1.28$. Rearranging for $x$, we get $x=\mu+z\sigma=9.64+( - 1.28)\times1.26=9.64-1.6128 = 8.0272\approx8.03$. But the given answer is $17.7$ which is incorrect.

Answer:

The above are the step - by - step solutions for each problem, but some of the given answers seem to have errors in calculation or approximation methods used.