Question # 1. Find each of the requested values for a population with a mean of ? = 40, and a standard deviation of ? = 8 A. What is the z-score corresponding to

Modeling data distributions
ANSWERED 1. Find each of the requested values for a population with a mean of $$? = 40$$, and a standard deviation of $$? = 8$$ A. What is the z-score corresponding to $$X = 52?$$ B. What is the X value corresponding to $$z = - 0.50?$$ C. If all of the scores in the population are transformed into z-scores, what will be the values for the mean and standard deviation for the complete set of z-scores? D. What is the z-score corresponding to a sample mean of $$M=42$$ for a sample of $$n = 4$$ scores? E. What is the z-scores corresponding to a sample mean of $$M= 42$$ for a sample of $$n = 6$$ scores? 2. True or false: a. All normal distributions are symmetrical b. All normal distributions have a mean of 1.0 c. All normal distributions have a standard deviation of 1.0 d. The total area under the curve of all normal distributions is equal to 1 3. Interpret the location, direction, and distance (near or far) of the following zscores: $$a. -2.00 b. 1.25 c. 3.50 d. -0.34$$ 4. You are part of a trivia team and have tracked your team’s performance since you started playing, so you know that your scores are normally distributed with $$\mu = 78$$ and $$\sigma = 12$$. Recently, a new person joined the team, and you think the scores have gotten better. Use hypothesis testing to see if the average score has improved based on the following 8 weeks’ worth of score data: $$82, 74, 62, 68, 79, 94, 90, 81, 80$$. 5. You get hired as a server at a local restaurant, and the manager tells you that servers’ tips are $42 on average but vary about $$12 (\mu = 42, \sigma = 12)$$. You decide to track your tips to see if you make a different amount, but because this is your first job as a server, you don’t know if you will make more or less in tips. After working 16 shifts, you find that your average nightly amount is$44.50 from tips. Test for a difference between this value and the population mean at the $$\alpha = 0.05$$ level of significance. 2020-10-24

(A) Obtain the z-score corresponding to $$X = 52$$. The z-score corresponding to $$X = 52$$ is obtained below as follows, From the information given that, Let X denotes the random variable with the population mean 40 and the standard deviation of 8 Thatis, $$\mu = 40, \sigma = 8$$ The required value is, $$z=\frac{X-\mu}{\sigma}$$
$$=\frac{52-40}{8}= \frac{12}{8}$$
$$= 1.50$$ Thus, the value of the z-score corresponding to $$X = 52 \text{ is } 1.50.$$ (B) Obtain the X value corresponding to $$z = –0.50$$. The X value corresponding to $$z = –0.50$$ is obtained below as follows: The required value is, $$z = X - \frac{\mu}{\sigma}$$
$$-0.50 = \frac{X-40}{8}$$
$$-0.50 \times 8.0 = X-40$$
$$-4.00 = X - 40$$
$$X = 40 - 4$$
$$= 36$$ Thus, the X value corresponding to $$z = –0.50 \text{ is } 36$$ It is clear that standard normal distribution represents a nommal curve with mean 0 and standard deviation 1 if the scores in the population are transformed into z-scores with the parameters involved in a nommal distribution are mean $$(\mu)$$ and standard deviation $$(\sigma)$$. Determine the value of mean if all of the scores in the population are transformed into z-scores The value of mean if all of the scores in the population are transformed into z-scores is 0. Determine the value of standard deviation if all of the scores in the population are transformed into z-scores. The value of standard deviation if all of the scores in the population are transformed into z-scores is 1.