Answer:
a) 0.011 = 1.1% probability that the sample mean hardness for a random sample of 17 pins is at least 51
b) 0.0001 = 0.1% probability that the sample mean hardness for a random sample of 45 pins is at least 51
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean and standard deviation
, the zscore of a measure X is given by:
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this problem, we have that:
(a) If the distribution is normal, what is the probability that the sample mean hardness for a random sample of 17 pins is at least 51?
Here
This probability is 1 subtracted by the pvalue of Z when X = 51. So
By the Central Limit Theorem
has a pvalue of 0.9890
1 - 0.989 = 0.011
0.011 = 1.1% probability that the sample mean hardness for a random sample of 17 pins is at least 51
(b) What is the (approximate) probability that the sample mean hardness for a random sample of 45 pins is at least 51?
Here
has a pvalue of 0.9999
1 - 0.9999 = 0.0001
0.0001 = 0.1% probability that the sample mean hardness for a random sample of 45 pins is at least 51
Solution :
Give that a shark can chase prey about 30 miles per hour.
1 hour = 60 minute
by using unitary method, we will convert miles per hour into miles per minute
in 60 minutes the shark can chase the prey 30 miles
In 1 minute the shark can chase
Hence the shark can chase the prey about 0.5 miles per minute.
We are asked to determine the correlation factor "r" of the given table. To do that we will first label the column for "Quality" as "x" and the column for "Easiness" as "y". Like this:
Now, we create another column with the product of "x" and "y". Like this:
Now, we will add another column with the squares of the values of "x". Like this:
Now, we add another column with the squares of the values of "y":
Now, we sum the values on each of the columns:
Now, to get the correlation factor we use the following formula:
Where:
Now we substitute the values, we get:
Solving the operations:
Therefore, the correlation factor is 0.858. If the correlation factor approaches the values of +1, this means that there is a strong linear correlation between the variables "x" and "y" and this correlation tends to be with a positive slope.
