Thickness measurements of ancient prehistoric Native American pot shards discovered in a Hopi village are approximately normally distributed, with a mean of 4.9millimeters (mm) and a standard deviation of 1.5 mm. For a randomly found shard, find the following probabilities. (Round your answers to four decimal places.)
(a) the thickness is less than 3.0 mm
(b) the thickness is more than 7.0 mm
(c) the thickness is between 3.0 mm and 7.0 mm
a) Find the z-score that corresponds to a thickness of 3.0 mm:
z = (x - μ)/σ
z = (3.0 - 4.9)/1.5
z ~ -1.27
So the probability that x < 3.0, or P(X<3.0) is the same as the probability that z < -1.27.
Consult a z-score table for this probability: find -1.2 in the left column and 0.07 in the top row. These intersect at 0.10204, or ~10.2%.
b) Use the same process, except this time the probability will be 1 - P(X<7.0).
c) This time, the probability will be P(X<7.0) - P(X<3.0). Should be quick as you found these numbers in parts a) and b).
The 5.3 millimeters and standard deviation of 1.5 mm, the thickness of less tan 3.0 _——— .
Assume that x has normal distribution with the specified mean and standard deviation. Find the indicated probability ? u=112;o=16. P(x_> 90) =?
The answer for "A" is -2.3333 because you subtract 3.0-5.1 which equals -2.1 and then divide it by 0.9 which equals -2.3333
(a) Less than 3.0 mm? Wow, those pot shards must have been very delicate! Let's use some clown math to find the probability.
First, we need to standardize the value of 3.0 mm using the formula:
z = (x - μ) / σ,
where x is the value, μ is the mean, and σ is the standard deviation. Plugging in the values, we get:
z = (3.0 - 4.9) / 1.5 = -1.2667.
Now, we can find the probability using a standard normal distribution table or calculator.
P(thickness < 3.0 mm) = P(z < -1.2667) = 0.1026.
So, the probability of finding a pot shard with a thickness less than 3.0 mm is approximately 0.1026.
(b) More than 7.0 mm? Phew, those would be some chunky pot shards! Let's use the same method as before.
Standardizing the value of 7.0 mm gives us:
z = (7.0 - 4.9) / 1.5 = 1.4000.
Now, let's find the probability.
P(thickness > 7.0 mm) = P(z > 1.4000)
Using clown magic (or a standard normal distribution table), we find that P(z > 1.4000) ≈ 0.0808.
So, the probability of finding a pot shard with a thickness more than 7.0 mm is approximately 0.0808.
(c) Between 3.0 mm and 7.0 mm? Now we're talking pottery in the goldilocks zone! To find this probability, we need to find the area under the curve between these two values.
First, let's standardize both values:
For 3.0 mm, z1 = (3.0 - 4.9) / 1.5 = -1.2667.
For 7.0 mm, z2 = (7.0 - 4.9) / 1.5 = 1.4000.
Now, the probability can be calculated as:
P(3.0 mm < thickness < 7.0 mm) = P(-1.2667 < z < 1.4000).
Using clown logic (or a standard normal distribution table), we find P(-1.2667 < z < 1.4000) ≈ 0.8132.
So, the probability of finding a pot shard with a thickness between 3.0 mm and 7.0 mm is approximately 0.8132.
Remember, these answers are rounded to four decimal places, but don't worry, they're as accurate as juggling chainsaws!
To find the probabilities for the given thickness measurements, we can use the standard normal distribution table or calculate the z-scores for each scenario and use those to find the probabilities. Let's calculate the probabilities step by step:
(a) The thickness is less than 3.0 mm:
To calculate this probability, we need to find the area under the normal curve to the left of 3.0 mm. First, we need to calculate the z-score for 3.0 mm using the formula:
z = (x - μ) / σ
where
x = 3.0 mm (value we want to find the probability for)
μ = 4.9 mm (mean)
σ = 1.5 mm (standard deviation)
Plugging in the values, we get:
z = (3.0 - 4.9) / 1.5
z = -1.27
Now, we can look up the cumulative probability associated with the z-score -1.27 in the standard normal distribution table. The value is approximately 0.1026.
Therefore, the probability that the thickness is less than 3.0 mm is approximately 0.1026.
(b) The thickness is more than 7.0 mm:
Similarly, we need to calculate the z-score for 7.0 mm:
z = (7.0 - 4.9) / 1.5
z = 1.40
Again, looking up the cumulative probability associated with the z-score 1.40 in the standard normal distribution table, we find approximately 0.9192.
Therefore, the probability that the thickness is more than 7.0 mm is approximately 0.0808.
(c) The thickness is between 3.0 mm and 7.0 mm:
To find this probability, we need to find the area under the normal curve between 3.0 mm and 7.0 mm. Since both values are already given in millimeters, we don't need to calculate z-scores separately.
Using the standard normal distribution table, we find the cumulative probabilities associated with z = -1.27 and z = 1.40, which are 0.1026 and 0.9192, respectively.
The probability of the thickness being between 3.0 mm and 7.0 mm can be calculated as the difference between the two probabilities:
P(3.0 mm < thickness < 7.0 mm) = P(thickness < 7.0 mm) - P(thickness < 3.0 mm)
= 0.9192 - 0.1026
= 0.8166
Therefore, the probability that the thickness is between 3.0 mm and 7.0 mm is approximately 0.8166.
the z-score of a sample is the number of standard deviations its thickness is from the mean thickness
calculate the z-scores , then use a z-score table to find the probabilities