Questions: (a) Can we use the normal distribution to compute probabilities of length of unemployment of one individual in this case? Yes, we can use normal distribution to compute probabilities of length of unemployment of one individual because we are told that the distribution of length of unemployment of one individual is normally distributed. No, we cannot use normal distribution to compute probabilities of length of unemployment of one individual because the distribution of length of unemployment of one individual is unknown. No, we cannot use normal distribution compute probabilities of length of unemployment of one individual because we don't know the standard deviation of the length of unemployment of one individual. Yes, we can use normal distribution to compute probabilities of length of unemployment of one individual because the distribution of length of unemployment of one individual is always normally distributed. (b) Can we use the normal distribution to compute probabilities of length of unemployment of the average of 20 unemployed individuals in this case? Yes, we can use the normal distribution to compute probabilities of length of unemployment of the average of 20 unemployed individuals because the sample size 20 is large enough. No, we cannot use the normal distribution to compute probabilities of length of unemployment of the average of 20 unemployed individuals because the sample size 20 is not large enough. Yes, we can use the normal distribution to compute probabilities of length of unemployment of the average of 20 unemployed individuals because the average length of unemployment is always normally distributed. No, we cannot use the normal distribution to compute probabilities of length of unemployment of the average of 20 unemployed individuals because we can't compute the standard error in this case. (c) State the Central Limit Theorem. If a random sample of size n is taken from any population, the distribution of the sample standard deviation becomes normal as the sample size increases. If a random sample of size n is taken from any population, the distribution of the sample mean is normal regardless of the sample size. If a random sample of size n is taken from any population, the distribution of the sample mean becomes normal as the sample size increases. If a random sample of size n is taken from any population, the distribution of the sample median becomes normal as the sample size increases.

(a) Can we use the normal distribution to compute probabilities of length of unemployment of one individual in this case? Yes, we can use normal distribution to compute probabilities of length of unemployment of one individual because we are told that the distribution of length of unemployment of one individual is normally distributed. No, we cannot use normal distribution to compute probabilities of length of unemployment of one individual because the distribution of length of unemployment of one individual is unknown. No, we cannot use normal distribution compute probabilities of length of unemployment of one individual because we don't know the standard deviation of the length of unemployment of one individual. Yes, we can use normal distribution to compute probabilities of length of unemployment of one individual because the distribution of length of unemployment of one individual is always normally distributed. (b) Can we use the normal distribution to compute probabilities of length of unemployment of the average of 20 unemployed individuals in this case? Yes, we can use the normal distribution to compute probabilities of length of unemployment of the average of 20 unemployed individuals because the sample size 20 is large enough. No, we cannot use the normal distribution to compute probabilities of length of unemployment of the average of 20 unemployed individuals because the sample size 20 is not large enough. Yes, we can use the normal distribution to compute probabilities of length of unemployment of the average of 20 unemployed individuals because the average length of unemployment is always normally distributed. No, we cannot use the normal distribution to compute probabilities of length of unemployment of the average of 20 unemployed individuals because we can't compute the standard error in this case. (c) State the Central Limit Theorem. If a random sample of size n is taken from any population, the distribution of the sample standard deviation becomes normal as the sample size increases. If a random sample of size n is taken from any population, the distribution of the sample mean is normal regardless of the sample size. If a random sample of size n is taken from any population, the distribution of the sample mean becomes normal as the sample size increases. If a random sample of size n is taken from any population, the distribution of the sample median becomes normal as the sample size increases.
Transcript text: (a) Can we use the normal distribution to compute probabilities of length of unemployment of one individual in this case? Yes, we can use normal distribution to compute probabilities of length of unemployment of one individual because we are told that the distribution of length of unemployment of one individual is normally distributed. No, we cannot use normal distribution to compute probabilities of length of unemployment of one individual because the distribution of length of unemployment of one individual is unknown. No, we cannot use normal distribution compute probabilities of length of unemployment of one individual because we don't know the standard deviation of the length of unemployment of one individual. Yes, we can use normal distribution to compute probabilities of length of unemployment of one individual because the distribution of length of unemployment of one individual is always normally distributed. (b) Can we use the normal distribution to compute probabilities of length of unemployment of the average of 20 unemployed individuals in this case? Yes, we can use the normal distribution to compute probabilities of length of unemployment of the average of 20 unemployed individuals because the sample size 20 is large enough. No, we cannot use the normal distribution to compute probabilities of length of unemployment of the average of 20 unemployed individuals because the sample size 20 is not large enough. Yes, we can use the normal distribution to compute probabilities of length of unemployment of the average of 20 unemployed individuals because the average length of unemployment is always normally distributed. No, we cannot use the normal distribution to compute probabilities of length of unemployment of the average of 20 unemployed individuals because we can't compute the standard error in this case. (c) State the Central Limit Theorem. If a random sample of size $n$ is taken from any population, the distribution of the sample standard deviation becomes normal as the sample size increases. If a random sample of size $n$ is taken from any population, the distribution of the sample mean is normal regardless of the sample size. If a random sample of size $n$ is taken from any population, the distribution of the sample mean becomes normal as the sample size increases. If a random sample of size $n$ is taken from any population, the distribution of the sample median becomes normal as the sample size increases.
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Solution

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Solution Steps

Step 1: Normal Distribution for Individual Unemployment Length

We can use the normal distribution to compute probabilities of the length of unemployment for one individual because the distribution of length of unemployment is assumed to be normally distributed.

Step 2: Normal Distribution for Sample Mean

For the average length of unemployment of a sample of 20 unemployed individuals, we can also use the normal distribution. This is justified by the Central Limit Theorem, which states that the distribution of the sample mean approaches normality as the sample size increases, and in this case, \( n = 20 \) is sufficiently large.

Step 3: Central Limit Theorem

The Central Limit Theorem states that if a random sample of size \( n \) is taken from any population, the distribution of the sample mean becomes normal as the sample size increases. Thus, we can apply this theorem to our sample of unemployed individuals.

Step 4: Probability Calculation

We calculated the probability that the sample mean length of unemployment falls between 28 weeks and 31 weeks. The Z-scores for these bounds were computed as follows:

  • For \( 28 \) weeks: \( Z_{start} = -0.6957 \)
  • For \( 31 \) weeks: \( Z_{end} = 0.795 \)

Using the cumulative distribution function \( \Phi \), we find: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(0.795) - \Phi(-0.6957) = 0.5434 \]

Final Answer

The answers to the sub-questions are:

  • (a) Yes, we can use the normal distribution.
  • (b) Yes, we can use the normal distribution.
  • (c) If a random sample of size \( n \) is taken from any population, the distribution of the sample mean becomes normal as the sample size increases.

The probability that the sample mean is between 28 and 31 weeks is \( P = 0.5434 \).

Thus, the final answer is: \[ \boxed{P = 0.5434} \]

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