Rosalina wants to save $700 for the new smart tv write and equation to determine how many weeks (x) it will take for rosalina to

Answer:

a) There is an 18.94% chance that the sample mean of the amount purchased will be at least 12 gallons.

b) There is an 81.06% chance that the total gasoline purchased will not exceed 600 gallons.

c) The estimated value for the 95th percentile of the total consumption by 50 randomly chosen customers is 621.5 gallons.

Step-by-step explanation:

The solution to this query involves applying the normal probability distribution and the central limit theorem.

Normal probability distribution

Issues involving normally distributed samples can be addressed using the z-score formula.

In a dataset characterized by mean and standard deviation , the z-score for a value X is expressed as:

The z-score indicates how many standard deviations a particular value is from the mean. After calculating the z-score, we reference the z-score table to find its corresponding p-value, which represents the probability that a measure is less than X, essentially giving us X's percentile. By subtracting the p-value from 1, we find the chance that the measure exceeds X.

Central Limit Theorem

The Central Limit Theorem posits that for a normally distributed variable X, with mean and standard deviation , the distribution of sample means with size n approximates a normal distribution characterized by mean and standard deviation .

Even when dealing with a skewed variable, the Central Limit Theorem remains applicable as long as n is no less than 30.

For sums, this theorem can likewise be employed, accompanied by mean and standard deviation .

In this scenario, we are given that:

a. For a group of 50 randomly selected customers, what is the estimated probability that the sample mean amount purchased is at least 12 gallons?

Here we have

This probability is derived from 1 minus the p-value of Z corresponding to X = 12.

According to the Central Limit Theorem

yields a p-value of 0.8106.

1 - 0.8106 = 0.1894

Therefore, there is an 18.94% chance that the sample mean amount purchased is at least 12 gallons.

b. For a group of 50 randomly selected customers, what is the estimated probability that the total amount of gasoline purchased does not exceed 600 gallons?

Regarding sums, we have

This probability equals the p-value of Z when X = 600. Hence,

displays a p-value of 0.8106.

Thus, there is an 81.06% chance that the total gasoline purchased will be 600 gallons or less.

c. What is the approximate figure for the 95th percentile regarding the total purchases by 50 randomly chosen customers?

This value corresponds to X when Z indicates a p-value of 0.95, which occurs at Z = 1.645.

The 95th percentile estimate for the total amount purchased by 50 randomly selected customers stands at 621.5 gallons.

The correct illustration is provided.

Explanation:

Utilizing a tool like Geogebra, commence by creating a line segment. Label the endpoints as C and D.

Next, draw the perpendicular bisector of the segment and denote the intersection with CD as B, then introduce a point A above this line.

Measure the distance from C to B and from B to D. Both distances will be equal.

Measure the length from A to B. If this distance is not equal to that from C to B, adjust A along line AB until the distances match.

With a compass and straightedge:

First, create segment CD and ensure the endpoints are labeled.

Adjust your compass to slightly more than half the distance between C and D. With it set at C, draw an arc above CD.

Using the same compass setting at D, draw another arc to intersect your first arc above CD. Mark the intersection as E.

Connect E to CD with a straightedge and label the intersection as B.

Set your compass to the distance from C to B. Position it on B and mark an arc on EB. Designate this intersection point as A.

Thus, AB will equal both CB and BD.

Let P(3) denote the probability of landing on 3 in a spin, and P(5) denote the probability of landing on 5. In probability terms, "AND" signifies multiplication while "OR" indicates addition. We aim to find the probability that the first number is "3" AND the second number is "5." Thus, we identify the individual probabilities and MULTIPLY them. The spinner has numbers ranging from 1 to 8, each appearing once. Therefore, since there is one instance of "3," we have P(3) = 1/8 and similarly P(5) = 1/8. Consequently, the overall probability of P(3 and 5) is 1/8 multiplied by 1/8, which equals 1/64.

Details provided:
Confidence level = 90%
Mean = 71 beats per minute
Standard deviation = 6 beats per minute

The formula for margin of error is z * δ / √n.

Where δ represents the population standard deviation and n is the sample size; z denotes the corresponding z-value.

For a 90% confidence level, the z-value is 1.645.

Thus, the margin of error is calculated as 1.645 * (6/√80) = 1.645 * (6/8.94) = 1.645 * 0.671 = 1.104.

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