Respuesta :
Answer:
[tex] \\ P(z>-2) = 0.97725[/tex] or P(x>49) is about 97.725% (or being less precise 97.5% using the empirical rule).
Step-by-step explanation:
We solve this question using the following information:
- We are dealing here with normally distributed data, that is "the frequency distribution of the life length data is known to be mound-shaped".
- The normal distribution is defined by two parameters: the population mean ([tex] \\ \mu[/tex]) and the population standard deviation ([tex] \\ \sigma[/tex]). In this case, we have that [tex] \\ \mu = 55[/tex] months, and [tex] \\ \sigma = 3[/tex] months.
- To find the probabilities, we have to use the standard normal distribution, which has [tex] \\ \mu = 0[/tex] and [tex] \\ \sigma = 1[/tex]. The probabilities for this distribution are collected in the standard normal table, available in Statistics books or on the Internet. We can also use statistics programs to find these probabilities.
- For most cases, we need to use the cumulative standard normal table, and for this we have to previously "transform" a raw score (x) into a z-score using the next formula: [tex] \\ z = \frac{x - \mu}{\sigma}[/tex] [1]. A z-score tells us the distance from the mean that a raw score is from it in standard deviations units. If this value is negative, the raw score is below the mean. Conversely, a positive value indicates that it is above the mean.
- The cumulative standard normal table is made for positive values of z. Since the normal distribution is symmetrical around the mean, we can find the negative values of z using this formula: [tex] \\ P(z<-a) = 1 - P(z<a) = P(z>a)[/tex] [2].
Having all this information, we can solve the question.
The percentage of the manufacturer's grade A batteries that will last more than 49 months
First Step: Use formula [1] to find the z-score of the raw score x = 49 months.
[tex] \\ z = \frac{49 - 55}{3}[/tex]
[tex] \\ z = \frac{-6}{3}[/tex]
[tex] \\ z = -2[/tex]
This means that the raw score is represented by a z-score of [tex] \\ z = -2[/tex], which tells us that it is two standard deviations below the population mean.
Second Step: Consult this value in the cumulative standard normal table for z = 2 and apply the formula [2] to find the corresponding probability.
For a z = 2, the probability is 0.97725.
Then
[tex] \\ P(z<-2) = 1 - P(z<2) = P(z>2)[/tex]
[tex] \\ P(z<-2) = 1 - 0.97725 = P(z>2)[/tex]
[tex] \\ P(z<-2) = 0.02275 = P(z>2)[/tex]
But we are not asked for P(z<-2) but for P(z>-2) = P(x>49). This probability is the complement of the previous result, that is
[tex] \\ P(z>-2) = 1 - P(z<-2)[/tex]
[tex] \\ P(z>-2) = 1 - 0.02275[/tex]
[tex] \\ P(z>-2) = 0.97725[/tex]
That is, the "percentage of the manufacturer's grade A batteries will last more than 49 months" is
[tex] \\ P(z>-2) = 0.97725[/tex] or about 97.725%
A graph below shows this result.
Notice that if we had used the 68-95-99.7 rule (also known as the empirical rule), that is, in a normal distribution, the interval between one standard deviation below and above the mean contains, approximately, 68% of the observations; the interval between two standard deviations below and above the mean contains, approximately, 95% of the observations; and the interval between three standard deviations below and above the mean contains, approximately, 99.7% of the observations, we could have concluded that 2.5 % of the manufacturer's grade A batteries will last less than 49 months, and, as a result, 1 - 0.025 = 0.975 or 97.5% will last more than 49 months.
We can conclude that with a less precise answer (but faster) because of the symmetry of the normal distribution, that is, 1 - 0.95 = 0.05. At both extremes we have 0.05/2 = 0.025 or 2.5% and we were asked for P(x>49) = P(z>-2) (see the graph below).
97.72% of the manufacturers grade A batteries will last more than 49 months
What is z score?
Z score is used to determine by how many standard deviations the raw score is above or below the mean.
It is given by:
z = (raw score - mean) / standard deviation
Mean = 55 months, standard deviation = 3 months
For x = 49:
z = (49 - 55)/3 = -2
P(z > -2) = 1 - P(z < -2) = 1 - 0.0228 = 0.9772
97.72% of the manufacturers grade A batteries will last more than 49 months
Find out more on z score at: https://brainly.com/question/25638875
