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Statistics homework help?
Your company is producing special battery packs for the most popular toy during the holiday season. The life span of the battery pack is known to be Normally distributed with a mean of 250 hours and a standard deviation of 20 hours.

1) What is the percentage of battery packs that last longer than 260 hours?
Give your answer to two decimal places.

2)If a simple random sample of four battery packs is selected from your company and we assume that their life spans are independent, the probability that they all last longer than 260 hours is ________.
Give your answer to 5 decimal places.

3)If a simple random sample of four battery packs is selected from your company, what is the probability that the average lifetime of these four packs is longer than 260 hours (express answer to 4 decimal places)?

1. Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.

2. Same, except to get probability that all events occur, multiply the probabilities of the individual events.

3. Since you are now dealing with the mean of the 4 packs, the equation becomes:

Z = (score-mean)/SEm

SEm = SD/√n

Use same table.

75%

To solve these questions, we will use the properties of the normal distribution.

1) To find the percentage of battery packs that last longer than 260 hours, we need to calculate the area under the curve to the right of 260 hours.

To find this probability, we can use the z-score formula:

z = (x - μ) / σ

where x is the value for which we want to find the probability, μ is the mean of the distribution, and σ is the standard deviation.

In this case, x = 260 hours, μ = 250 hours, and σ = 20 hours.

Substituting the values into the formula:

z = (260 - 250) / 20
= 0.5

Now, we need to find the area to the right of the z-score of 0.5. We can use a standard normal distribution table or a calculator to find this probability.

Using a calculator or a standard normal distribution table:

P(Z > 0.5) = 1 - P(Z < 0.5)

The probability, rounded to two decimal places, is approximately 0.3085.

2) To find the probability that all four battery packs last longer than 260 hours, we can assume that their lifespans are independent and apply the multiplication rule for independent events.

Since the probability of each individual battery pack lasting longer than 260 hours is 0.3085 (as calculated above), the probability that all four packs last longer than 260 hours is:

P(all last longer than 260) = P(first lasts longer than 260) * P(second lasts longer than 260) * P(third lasts longer than 260) * P(fourth lasts longer than 260)

P(all last longer than 260) = (0.3085) * (0.3085) * (0.3085) * (0.3085)

The probability, rounded to 5 decimal places, is approximately 0.02967.

3) To find the probability that the average lifetime of these four packs is longer than 260 hours, we can use the central limit theorem. According to the central limit theorem, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population distribution.

The mean of the sampling distribution of the sample mean is equal to the population mean, which is 250 hours. The standard deviation of the sampling distribution, also known as the standard error of the mean, can be calculated using the formula:

σ(𝑚𝑒𝑎𝑛) = σ / sqrt(n)

where σ is the population standard deviation and n is the sample size.

In this case, σ = 20 hours and n = 4.

σ(𝑚𝑒𝑎𝑛) = 20 / sqrt(4)
= 10

To find the probability that the average lifetime of these four packs is longer than 260 hours, we need to calculate the z-score using the formula:

z = (x - μ(𝑚𝑒𝑎𝑛)) / σ(𝑚𝑒𝑎𝑛)

where x is the value for which we want to find the probability, μ(𝑚𝑒𝑎𝑛) is the mean of the sampling distribution (which is the same as the population mean), and σ(𝑚𝑒𝑎𝑛) is the standard deviation of the sampling distribution.

In this case, x = 260 hours, μ(𝑚𝑒𝑎𝑛) = 250 hours, and σ(𝑚𝑒𝑎𝑛) = 10 hours.

Substituting the values into the formula:

z = (260 - 250) / 10
= 1

Now, we need to find the area to the right of the z-score of 1.

Using a calculator or a standard normal distribution table:

P(Z > 1) = 1 - P(Z < 1)

The probability, rounded to 4 decimal places, is approximately 0.1587.

Therefore, the probability that the average lifetime of these four packs is longer than 260 hours is approximately 0.1587.

To solve these questions, we will use the properties of the normal distribution.

1) To find the percentage of battery packs that last longer than 260 hours, we need to find the area under the normal curve to the right of 260.

Step-by-step solution:
a) Standardize the value of 260 using the formula z = (x - mean) / standard deviation.
z = (260 - 250) / 20
z = 0.5

b) Lookup the standard normal distribution table or use a calculator to find the area to the right of 0.5.
The area to the right of 0.5 is 0.3085.

c) Convert the area to a percentage by multiplying by 100.
Percentage = 0.3085 * 100 ≈ 30.85%

So, the percentage of battery packs that last longer than 260 hours is approximately 30.85%.

2) To find the probability that all four battery packs last longer than 260 hours, we need to calculate the probability of each individual battery pack lasting longer than 260 hours and then multiply them together.

Step-by-step solution:
a) Calculate the probability of a single battery pack lasting longer than 260 hours using the z-score.
z = (260 - 250) / 20 = 0.5

b) Lookup the standard normal distribution table or use a calculator to find the probability of z > 0.5.
The probability is 0.3085.

c) Since the battery packs are assumed to be independent, we can multiply the probabilities together.
Probability of all four lasting longer than 260 hours = (0.3085)^4 ≈ 0.00389

So, the probability that all four battery packs last longer than 260 hours is approximately 0.00389.

3) To find the probability that the average lifetime of four battery packs is longer than 260 hours, we will use the Central Limit Theorem, which states that the distribution of sample means from a population with any distribution approaches a normal distribution as the sample size increases.

Step-by-step solution:
a) Since the sample size is only four, we can assume that the distribution of the average lifetime follows a normal distribution with the same mean (250 hours) but a smaller standard deviation.

b) The standard deviation of the sample mean can be calculated using the formula:
Standard deviation of sample mean = population standard deviation / sqrt(sample size)
Standard deviation of sample mean = 20 / sqrt(4) = 10

c) Standardize the value of 260 using the formula z = (x - mean) / standard deviation.
z = (260 - 250) / 10 = 1

d) Lookup the standard normal distribution table or use a calculator to find the probability of z > 1.
The probability is 0.1587.

So, the probability that the average lifetime of four battery packs is longer than 260 hours is approximately 0.1587.