To solve this problem, we need to use the Central Limit Theorem which states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed.

This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large (usually n 30). If the population is normal, then the theorem holds true even for samples smaller than 30. In this case, we have a sample size of 10, and we know the population is normally distributed, so we can use the Central Limit Theorem.

Here are the steps to solve the problem:

1. First, we need to find the standard deviation of the sample mean. The formula to find the standard deviation of the sample mean (often called the standard error) is σ/√n. Plugging in the given values, we get √40/√10 = √4 = 2.

2. Next, we need to convert the raw score of 42 to a z-score. The formula to find the z-score is (X - μ) / σ. Plugging in the given values, we get (42 - 40) / 2 = 1.

3. Finally, we need to find the probability that the z-score is less than or equal to 1. Looking up 1 in the z-table, we find that the probability is 0.8413.

So, P(X¯≤42) = 0.8413.

Question

To solve this problem, we need to use the Central Limit Theorem which states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed.

This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large (usually n > 30). If the population is normal, then the theorem holds true even for samples smaller than 30. In this case, we have a sample size of 10, and we know the population is normally distributed, so we can use the Central Limit Theorem.

Here are the steps to solve the problem:

1. First, we need to find the standard deviation of the sample mean. The formula to find the standard deviation of the sample mean (often called the standard error) is σ/√n. Plugging in the given values, we get √40/√10 = √4 = 2.

2. Next, we need to convert the raw score of 42 to a z-score. The formula to find the z-score is (X - μ) / σ. Plugging in the given values, we get (42 - 40) / 2 = 1.

3. Finally, we need to find the probability that the z-score is less than or equal to 1. Looking up 1 in the z-table, we find that the probability is 0.8413.

So, P(X¯≤42) = 0.8413.

Knowee AI · Accepted Answer

To solve this problem, we need to use the Central Limit Theorem which states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed.

This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large (usually n > 30). If the population is normal, then the theorem holds true even for samples smaller than 30. In this case, we have a sample size of 10, and we know the population is normally distributed, so we can use the Central Limit Theorem.

Here are the steps to solve the problem:

1. First, we need to find the standard deviation of the sample mean. The formula to find the standard deviation of the sample mean (often called the standard error) is σ/√n. Plugging in the given values, we get √40/√10 = √4 = 2.

2. Next, we need to convert the raw score of 42 to a z-score. The formula to find the z-score is (X - μ) / σ. Plugging in the given values, we get (42 - 40) / 2 = 1.

3. Finally, we need to find the probability that the z-score is less than or equal to 1. Looking up 1 in the z-table, we find that the probability is 0.8413.

So, P(X¯≤42) = 0.8413.

6. Suppose the random variable X𝑋 is normally distributed with μ=40𝜇=40 and σ2=40𝜎2=40 . What is P(X¯≤42)P(𝑋¯≤42) if the sample size n𝑛 is equal to 10?Multiple choice 4 Question 3 0.6915 0.1587 0.8413 0.3085

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