Native American pot shards discovered in a hopi villiage are approximately normally distributed with a mean of 5.1 millimeters and a standard deviationof 0.9mm what is the probability that the thickness is (a)less than 3.0mm (b) more than 7.0 and (c) between 3.omm and 7.0mm

To find the probabilities in this scenario, we can use the concept of z-scores and the standard normal distribution. By converting the given values into z-scores, we can determine the corresponding probabilities using a standard normal distribution table or a calculator.

The formula to calculate the z-score is:

z = (x - μ) / σ

Where:
- x is the given value (thickness)
- μ is the mean thickness
- σ is the standard deviation of thickness

Let's calculate the probabilities for each case:

(a) Probability that the thickness is less than 3.0mm:
First, calculate the z-score:

z = (3.0 - 5.1) / 0.9
z = -2.333
(The negative sign indicates that it is on the left side of the mean)

Using a standard normal distribution table or a calculator, you can find that the cumulative probability for a z-score of -2.333 is approximately 0.0099. This means that the probability of the thickness being less than 3.0mm is about 0.0099, or 0.99%.

(b) Probability that the thickness is more than 7.0mm:
Again, calculate the z-score:

z = (7.0 - 5.1) / 0.9
z = 2.111
(The positive sign indicates that it is on the right side of the mean)

Using the same methods, you can find that the cumulative probability for a z-score of 2.111 is approximately 0.9821. This means that the probability of the thickness being more than 7.0mm is about 0.9821, or 98.21%.

(c) Probability that the thickness is between 3.0mm and 7.0mm:
To find this probability, we need to find the area under the curve between these two values.

First, calculate the z-score for 3.0mm:

z1 = (3.0 - 5.1) / 0.9
z1 = -2.333

Next, calculate the z-score for 7.0mm:

z2 = (7.0 - 5.1) / 0.9
z2 = 2.111

Now, using either a standard normal distribution table, a calculator, or subtracting probabilities, you can find the difference between the cumulative probabilities for these two z-scores:

P(3.0 < thickness < 7.0) = P(thickness < 7.0) - P(thickness < 3.0)

P(thickness < 7.0) = 0.9821 (from part b)
P(thickness < 3.0) = 0.0099 (from part a)

P(3.0 < thickness < 7.0) = 0.9821 - 0.0099 = 0.9722, or 97.22%

Therefore, the probability that the thickness is (a) less than 3.0mm is approximately 0.99%, (b) more than 7.0mm is approximately 98.21%, and (c) between 3.0mm and 7.0mm is approximately 97.22%.