Question

A leading magazine (like Barron's) reported at one time that the average number of weeks an individual is unemployed is 28 weeks. Assume that the length of unemployment is normally distributed with population mean of 28 weeks and the population standard deviation of 9 weeks. Suppose you would like to select a random sample of 35 unemployed individuals for a follow-up study. Round the answers of following questions to 4 decimal places. What is the distribution of X? X ~ N What is the distribution of ¯x? ¯x ~ N What is the probability that one randomly selected individual found a job more than 30 weeks? For 35 unemployed individuals, find the probability that the average time that they found the next job is more than 30 weeks. For part d), is the assumption of normal necessary? yes or no

Answer 1

The distribution of the length of unemployment time for an individual (X) is normally distributed, and so is the distribution of the sample mean with a reduced standard deviation. The probabilities for an individual and for the sample mean exceeding 30 weeks need to be calculated using their respective z-scores and standard normal distribution. Normality assumption is important for smaller sample sizes, but for larger ones, the Central Limit Theorem provides an approximation.

Step-by-step explanation:

The random variable X represents the length of unemployment time for an individual, which is normally distributed with a mean (μ) of 28 weeks and a standard deviation (σ) of 9 weeks. For a sample of 35 individuals, the distribution of the sample mean is also normally distributed with the same mean (μ = 28 weeks) but a smaller standard deviation equal to σ/√n (9 weeks/√35).

To find the probability that one randomly selected individual found a job after more than 30 weeks, we calculate the z-score for 30 weeks and use the standard normal distribution to find the probability. For 35 unemployed individuals, to find the probability that the sample mean time to find the next job is more than 30 weeks, we again find the z-score using the distribution of the sample mean and refer to the standard normal distribution.

For part d), the assumption of normality is necessary for the exact calculation of probabilities when the sample size is small. However, due to the Central Limit Theorem, for, if the sample size is large (typically n > 30), the sample mean distribution would be approximately normal regardless of the population distribution.

Answer 2

The distribution of the length of unemployment (X) is normal, with
`\(X \sim N(28, 9^2)\)`. The distribution of the sample mean
`(\(\bar{x}\))` for a sample of 35 unemployed individuals is also normal, with
`\(\bar{x} \sim N(28, (9)/(√(35)))\)`. The probability that a randomly selected individual found a job after more than 30 weeks is given by
`\(P(X > 30)\)`, and for 35 unemployed individuals, the probability that the average time they found the next job is more than 30 weeks is given by
`\(P(\bar{x} > 30)\)`. The assumption of normality is necessary for these calculations.

Step-by-step explanation:

The distribution of the length of unemployment (X) is described by
`\(X \sim N(28, 9^2)\)`, indicating a normal distribution with a mean
`(\(\mu\))` of 28 weeks and a standard deviation
`(\(\sigma\))` of 9 weeks. Similarly, the distribution of the sample mean
`(\(\bar{x}\))` for a sample of 35 unemployed individuals is normal with a mean of 28 weeks
`(\(\mu_{\bar{x}} = \mu = 28\))` and a standard deviation
`(\(\sigma_{\bar{x}}\))` calculated as
`\((\sigma)/(√(n))\)` where
`\(n\)` is the sample size (35 in this case). Therefore,
`\(\bar{x} \sim N(28, (9)/(√(35)))\)`.

To find the probability that a randomly selected individual found a job after more than 30 weeks
`(\(P(X > 30)\))`, we use the standard normal distribution or z-score. Calculating the z-score for this scenario, we find the corresponding probability. Similarly, for 35 unemployed individuals, the probability that the average time they found the next job is more than 30 weeks
`(\(P(\bar{x} > 30)\))` involves using the distribution of the sample mean.

The assumption of normality is necessary for these probability calculations as the questions involve normal distribution properties. The Central Limit Theorem supports the normal approximation for the sample mean when the sample size is sufficiently large, ensuring accurate probability estimates based on normal distribution characteristics.