Question

The following problem uses Central Limit Theorem (finding the probability that the mean of one randpm group from all nossible groups of the same size from the population): Use this information to complete questions at numbers 1.4 below: Pulse rates of adult women are normally distributed with a mean of 74 beats per minute and a standard deviation of 12.5 beats per minute. 1. If 16 women are randomly selected from all possible groups of 16 women in the population, what is the probability their mean heart rate is greater than 80.25 bpm? Show the formula to Find the value of z:z= Include the probability notation to Find: P(x−bar>80.25)= 2. What is different in the probability notation when Central Limit Theorem is used? 3. What calculation(s) are done differently when Central Limit Theorem is used? 4. Why can we use Central Limit Theorem even though the sample size is less than 30 ?

Answer 1

The largest volume V for the box is given by
`\( V = x(3-2x)^2 \)`ft³, with
`\( x = (3)/(4) \)` maximizing the volume.

This expression derives from the problem constraints, where a square piece of 3 ft wide cardboard is used to construct an open-top box by cutting out squares from each corner, resulting in a maximum volume when
`\( x = (3)/(4) \)`.

Step-by-step explanation:

To determine the largest volume, we begin by understanding the problem's constraints. A square piece of cardboard, 3 ft wide, is used to construct an open-top box by cutting out squares from each corner and bending up the sides.

Let ( x ) be the length of the side of the square being cut out. The dimensions of the resulting box are (3-2x) by (3-2x) by ( x ). The volume V is calculated as the product of the length, width, and height, yielding
`\( V = x(3-2x)^2 \)`.

Maximizing V involves finding critical points by taking the derivative, setting it equal to zero, and solving for x . The solution,
`\( x = (3)/(4) \)`, represents the optimal value for x , providing the maximum volume of the box.

Answer 2

To find the probability of the mean heart rate of a sample of 16 women being greater than 80.25 bpm, we can use the Central Limit Theorem. The probability notation is P(Z > z), where Z is a standard normal random variable. When using the Central Limit Theorem, we calculate the z-score using the formula z = (x-bar - μ) / (σ / sqrt(n)).

Step-by-step explanation:

The probability that the mean heart rate of a random sample of 16 women is greater than 80.25 bpm can be found using the Central Limit Theorem. To find the value of z, we use the formula z = (x‑bar - μ) / (σ / sqrt(n)), where x‑bar is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size. In this case, x‑bar = 80.25 bpm, μ = 74 bpm, σ = 12.5 bpm, and n = 16.

The probability notation to find P(x‑bar > 80.25) is P(Z > z), where Z is a standard normal random variable with a mean of 0 and a standard deviation of 1. We need to calculate the z‑score using the formula z = (x‑bar - μ) / (σ / sqrt(n)). Plugging in the values, we get z = (80.25 - 74) / (12.5 / sqrt(16)) = 2.34.

Using a standard normal distribution table or a statistical software, we can find the probability P(Z > 2.34). The probability notation can vary slightly when using the Central Limit Theorem, but the concept remains the same. Instead of using specific values of the population, we use the mean and standard deviation of the sample.

When using the Central Limit Theorem, the calculation of the z-score is done differently. Instead of using the population mean and standard deviation, we use the sample mean and standard deviation. This allows us to estimate the probability of an event occurring in the population based on a sample.

The Central Limit Theorem can be used even when the sample size is less than 30 because it states that for a random sample of any size taken from a population with a finite mean and standard deviation, the distribution of the sample mean approaches a normal distribution as the sample size increases.