The Z score for a normal distribution at the  percentile is  So  , which can be found on the normal distribution table. The mass of tomatoes in the  percentile of all tomatoes  is  standard deviations below the mean, so the mass is .

First, we use our normal distribution table to find a p-value for a z-score greater than 0.50.

Our table tells us the probability is approximately,

.

Next we use our normal distribution table to find a p-value for a z-score greater than 1.23.

Our table tells us the probability is approximately,

.

We then subtract the probability of z being greater than 0.50 from the probability of z being less than 1.23 to give us our answer of,

.

First, we use the table to look up a p-value for z > -1.22.

This gives us a p-value of,

.

Next, we use the table to look up a p-value for z > 1.59.

This gives us a p-value of,

.

Finally we subtract the probability of z being greater than -1.22 from the probability of z being less than 1.59 to arrive at our answer of,

.

When we get this type of problem, first we need to calculate a z-score that we can use in our table.

To do that, we use our z-score formula:

where,

Plugging into the equation we get:

We then use our table to look up a p-value for z > 0.9. Since we want to calculate the probability of students who earned a higher score than Gabbie we need to subtract the P(z<0.9) to get our answer.

The two main parameters of the normal distribution are  and .  is a location parameter which determines the location of the peak of the normal distribution on the real number line.  is a scale parameter which determines the concentration of the density around the mean. Larger 's lead the normal to spread out more than smaller 's. 

The mean determines where the normal distribution lies on the real number line, while the variance determines the spread of the distribution. 

A normal distribution is one in which the values are evenly distributed both above and below the mean.  A population has a precisely normal distribution if the mean, mode, and median are all equal.  For the population of 3,4,5,5,5,6,7, the mean, mode, and median are all 5.

In a normal distribution, the number of values within one positive standard deviation of the mean is equal to the number of values within one negative standard deviation of the mean.  The reason for this is that the values below the population mean exactly parallel the values above the mean.

2) and 3) are true by definition of the Empirical Rule - also known as the 68-95-99.7 Rule. Using our information with mean of 100 and a standard deviation of 5 we can create a bell curve with 100 in the middle. One standard deviation out from the mean would give us a range from 95 to 105 and would be in our 68% section. If we go two standard deviations out from 100 we would get the range 90 to 110 thus lying in the 95% section. Lastly, when we go out 3 standard deviations we get a range of 85 to 115 thus falling within the 99.7% section.

1) can be deduced as true because it means that 68% of observations are between 95 and 105, and removing one of those bounds (namely, the upper one) adds the 16%, since , of observations larger than 105 leading to 68 + 16 = 84% of observations greater than 95.

A normal distrubtion is completely symmetrical and has not outliers.  This means that the mean is not pulled to either side by outliers and should lie directly in the middle along with the median (the middle number of the distribution).