# For a variable that is normally distributed with a mean of 50 and a standard deviation of 5, approximately _? of the scores are between _?. A)95%, 45 and 55 B)95%, 40 and 60 C)68%, 40 and 60 D)95%, 35 and 65 E)99%, 40 and 60 Question
Normal distributions For a variable that is normally distributed with a mean of 50 and a standard deviation of 5, approximately _? of the scores are between _?.
A)95%, 45 and 55
B)95%, 40 and 60
C)68%, 40 and 60
D)95%, 35 and 65
E)99%, 40 and 60 2020-11-06
Step 1
Empirical rule:
For symmetric or normal distributions,
68% of values fall within one standard deviation from the mean. That is, (m – s, m + s).
95% of values fall within two standard deviations from the mean. That is, (m – 2*s, m + 2*s).
7% of values fall within three standard deviations from the mean. That is, (m – 3*s, m + 3*s).
Step 2
Calculations:
The random variable x follows normal distribution with mean 50 and standard deviation 5.
That is, m = 50 and s = 5.
The calculation based on empirical rule is given below:
One $$\displaystyle\sigma$$ limits:
$$\displaystyle{\left(\mu\pm\sigma\right)}={\left({5}--{5},{50}+{5}\right)}={\left({45},{55}\right)}$$
68% of scores are between 45 and 55.
Two sigma limits:
$$\displaystyle{\left(\mu\pm{2}\sigma\right)}={\left({50}-{2}\times{5},{50}+{2}\times{5}\right)}={\left({40},{60}\right)}$$
95% of scores are between 40 and 60.
Three $$\displaystyle\sigma$$ limits:
$$\displaystyle{\left(\mu\pm{3}\sigma\right)}={\left({50}-{3}\times{5},{50}+{3}\times{5}\right)}={\left({35},{65}\right)}$$
99% of scores are between 35 and 65.
Step 3
Obtain the correct option:
From the given 5 options, option (B) states that 95% of the scores lie between 40 and 60.
Hence, option (B) is correct.
Step 4
Option (B) is correct.

### Relevant Questions 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. An automatic machine in a manufacturing process is operating properly if the lengths of an important subcomponent are normally distributed with a mean of 116 cm and a standard deviation of 4.8 cm.
A. Find the probability that one selected subcomponent is longer than 118 cm.
B. Find the probability that if 4 subcomponents are randomly selected, their mean length exceeds 118 cm.
C. Find the probability that if 4 are randomly selected, all 4 have lengths that exceed 118 cm. The article “Modeling Arterial Signal Optimization with Enhanced Cell Transmission Formulations presents a new method for timing traffic signals in heavily traveled intersections. The effectiveness of the new method was evaluated in a simulation study. In 50 simulations, the mean improvement in traffic flow in a particular intersection was 654.1 vehicles per hour, with a standard deviation of 311.7 vehicles per hour.
a) Find a $$\displaystyle{95}\%$$ confidence interval for the improvement in traffic flow due to the new system.
b) Find a $$\displaystyle{98}\%$$ confidence interval for the improvement in traffic flow due to the new system.
c) A traffic engineer states that the mean improvement is between 581.6 and 726.6 vehicles per hour. With what level of confidence can this statement be made?
d) Approximately what sample size is needed so that a $$\displaystyle{95}\%$$
confidence interval will specify the mean to within $$\displaystyle\pm\ {50}$$ vehicles per hour?
e) Approximately what sample size is needed so that a $$\displaystyle{98}\%$$ confidence
interval will specify the mean to within $$\displaystyle\pm\ {50}$$ vehicles per hour? 1. A researcher is interested in finding a 98% confidence interval for the mean number of times per day that college students text. The study included 144 students who averaged 44.7 texts per day. The standard deviation was 16.5 texts. a. To compute the confidence interval use a ? z t distribution. b. With 98% confidence the population mean number of texts per day is between and texts. c. If many groups of 144 randomly selected members are studied, then a different confidence interval would be produced from each group. About percent of these confidence intervals will contain the true population number of texts per day and about percent will not contain the true population mean number of texts per day. 2. You want to obtain a sample to estimate how much parents spend on their kids birthday parties. Based on previous study, you believe the population standard deviation is approximately $$\displaystyle\sigma={40.4}$$ dollars. You would like to be 90% confident that your estimate is within 1.5 dollar(s) of average spending on the birthday parties. How many parents do you have to sample? n = 3. You want to obtain a sample to estimate a population mean. Based on previous evidence, you believe the population standard deviation is approximately $$\displaystyle\sigma={57.5}$$. You would like to be 95% confident that your estimate is within 0.1 of the true population mean. How large of a sample size is required? The distribution of height for a certain population of women is approximately normal with mean 65 inches and standard deviation 3.5 inches. Consider two different random samples taken from the population, one of size 5 and one of size 85.
Which of the following is true about the sampling distributions of the sample mean for the two sample sizes?
Both distributions are approximately normal with mean 65 and standard deviation 3.5.
A
Both distributions are approximately normal. The mean and standard deviation for size 5 are both less than the mean and standard deviation for size 85.
B
Both distributions are approximately normal with the same mean. The standard deviation for size 5 is greater than that for size 85.
C
Only the distribution for size 85 is approximately normal. Both distributions have mean 65 and standard deviation 3.5.
D
Only the distribution for size 85 is approximately normal. The mean and standard deviation for size 5 are both less than the mean and standard deviation for size 85.
E The manager of the store in the preceding exercise calculated the residual for each point in the scatterplot and made a dotplot of the residuals.
The distribution of residuals is roughly Normal with a mean of $0 and standard deviation of$22.92.
The middle 95% of residuals should be between which two values? Use this information to give an interval of plausible values for the weekly sales revenue if 5 linear feet are allocated to the store's brand of men's grooming products. (a) The company's production equipment produces metal discs weighing 200 g. It should be noted that the weight of the discs corresponds to the normal distribution. To check machine consistency, 20 discs are randomly selected with an average weight of 205 g and a standard deviation of 7 g. What is the 99% confidence interval for the average weight of the selected discs?
(b) A company launched a new model of golf ball. It claimed that the driving distance is at least 300m. A sample of 20 balls yields a sample mean of 295m and sample standard deviation of 8m. It is assumed that the driving distance is normally distributed.
Conduct an appropriate hypothesis testing at the 0.05 level of significance. Is there any evidence that the average travel distance stated by the company is true? Assume that there is a sampling distribution of $$\displaystyle\overline{{{x}}}$$ of size 34 and they are selected at random from a normally distributed with a mean of 45 and a standard deviation of 6.2.
Find the probability that 1 randomly selected person has a value less than 40. True or False
1.The goal of descriptive statistics is to simplify, summarize, and organize data.
2.A summary value, usually numerical, that describes a sample is called a parameter.
3.A researcher records the average age for a group of 25 preschool children selected to participate in a research study. The average age is an example of a statistic.
4.The median is the most commonly used measure of central tendency.
5.The mode is the best way to measure central tendency for data from a nominal scale of measurement.
6.A distribution of scores and a mean of 55 and a standard deviation of 4. The variance for this distribution is 16.
7.In a distribution with a mean of M = 36 and a standard deviation of SD = 8, a score of 40 would be considered an extreme value.
8.In a distribution with a mean of M = 76 and a standard deviation of SD = 7, a score of 91 would be considered an extreme value.
9.A negative correlation means that as the X values decrease, the Y values also tend to decrease.
10.The goal of a hypothesis test is to demonstrate that the patterns observed in the sample data represent real patterns in the population and are not simply due to chance or sampling error. 