Help with number 2 and show how you got the answer

"A manufacturer of automobile batteries claims that the average length of life for its grade A battery is 55 months. Suppose the

STALIN [3.7K]

Answer:

$\\ P(z>-2) = 0.97725$ 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 ( $\\ \mu$ ) and the population standard deviation ( $\\ \sigma$ ). In this case, we have that $\\ \mu = 55$ months, and $\\ \sigma = 3$ months.
To find the probabilities, we have to use the standard normal distribution, which has $\\ \mu = 0$ and $\\ \sigma = 1$ . 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: $\\ z = \frac{x - \mu}{\sigma}$ [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: $\\ P(z$ [2].

Having all this information, we can solve the question.

<h3>The percentage of the manufacturer's grade A batteries that will last more than 49 months</h3>

First Step: Use formula [1] to find the z-score of the raw score x = 49 months.

$\\ z = \frac{49 - 55}{3}$

$\\ z = \frac{-6}{3}$

$\\ z = -2$

This means that the raw score is represented by a z-score of $\\ z = -2$ , 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

$\\ P(z$

$\\ P(z2)$

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

$\\ P(z>-2) = 1 - P(z$

$\\ P(z>-2) = 1 - 0.02275$

$\\ P(z>-2) = 0.97725$

That is, the "percentage of the manufacturer's grade A batteries will last more than 49 months" is

$\\ P(z>-2) = 0.97725$ 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).

6 0

3 years ago

A quadrilateral has diagonals that are congruent and bisect opposite pairs of

gtnhenbr [62]

If the diagonals of a quadrilateral are perpendicular, then it is a rhombus. False, diagonals don't have to be congruent or bisect each other. The diagonals of a rectangle bisect its angles. A kite with all consecutive angles congruent must be a square.

8 0

3 years ago

I need a thorough explanation, if not, only an answer with some steps would be fine.

Darya [45]

Answer:

$92.25

Step-by-step explanation:

If Diego spent $20.50 and Sandy spent 4.5x as much as Diego, that means that Sandy spent 20.5x4.5, which equals 92.25. Sandy spent $92.25

7 0

3 years ago

Out of 45 times at bat,Raul got 19 hits.Find Raul's batting average as a decimal rounded to the nearest thousandth

alukav5142 [94]

Answer:

0.422

Step-by-step explanation:

19/45 = 0.422

8 0

3 years ago

Read 2 more answers

he one‑sample t statistic from a sample of n = 23 observations for the two‑sided test of H 0 : μ = 15 versus H α : μ > 15 has

DedPeter [7]

Answer:

$t = 2.24$

The first step is calculate the degrees of freedom, on this case:

$df=n-1=23-1=22$

Since is a one side right tailed test the p value would be:

$p_v =P(t_{(22)}>2.24)=0.01776$

And for this case we can conclude that:

$0.01 < p_v < 0.025$

And we will reject the null hypothesis at $\alpha=0.025$ since $p_v < \alpha$

Step-by-step explanation:

Data given and notation

$\bar X$ represent the mean height for the sample

$s$ represent the sample standard deviation

$n=23$ sample size

$\mu_o =15$ represent the value that we want to test

t would represent the statistic (variable of interest)

$p_v$ represent the p value for the test (variable of interest)

State the null and alternative hypotheses.

We need to conduct a hypothesis in order to check if the mean is higher than 15, the system of hypothesis would be:

Null hypothesis: $\mu \leq 15$

Alternative hypothesis: $\mu > 15$

If we analyze the size for the sample is > 30 but we don't know the population deviation so is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:

$t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}$ (1)

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".

Calculate the statistic

For this case the statistic is given:

$t = 2.24$

P-value

The first step is calculate the degrees of freedom, on this case:

$df=n-1=23-1=22$

Since is a one side right tailed test the p value would be:

$p_v =P(t_{(22)}>2.24)=0.01776$

And for this case we can conclude that:

$0.01 < p_v < 0.025$

And we will reject the null hypothesis at $\alpha=0.025$ since $p_v < \alpha$

5 0

3 years ago