# Find the specified probability and interpret the results. Use technology to determine the likelihood of a challenge. For a particular week, the average gasoline price in California was $4.117 per gallon. A random sample of 38 filling stations is selected from this population. What is the likelihood that the average sample price this week was between$ 4.128 and $4.143? Assume sigma =$0.049
Find the specified probability and interpret the results. Use technology to determine the likelihood of a challenge. For a particular week, the average gasoline price in California was $4.117 per gallon. A random sample of 38 filling stations is selected from this population. What is the likelihood that the average sample price this week was between$ 4.128 and $4.143? Assume $$\sigma = 0.049$$ ## Answers (1) 2020-10-26 The sample mean is normally distibuted with mean mu and standard deviation $$\sigma/\sqrt{n}$$. the z-score is the value decreased by the mean, divided by the standard deviation: $$z=\frac{x-\mu}{\sigma/\sqrt{n}}=\frac{4.128-4.117}{0.049/\sqrt{38}}=1.38$$ $$z=\frac{x-\mu}{\sigma/\sqrt{n}}=\frac{4.143-4.117}{0.049/\sqrt{38}}=3.27$$ Determine the corresponding probability using table 4: $$P(4.128 < X < 84.143) = P(Z<3.27)-P(Z<1.38) = 0.9995-0.9162=0.0833=8.33 \%$$ ### Relevant Questions asked 2021-05-05 A random sample of $$n_1 = 14$$ winter days in Denver gave a sample mean pollution index $$x_1 = 43$$. Previous studies show that $$\sigma_1 = 19$$. For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$. Previous studies show that $$\sigma_2 = 13$$. Assume the pollution index is normally distributed in both Englewood and Denver. (a) State the null and alternate hypotheses. $$H_0:\mu_1=\mu_2.\mu_1>\mu_2$$ $$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$ $$H_0:\mu_1=\mu_2.\mu_1<\mu_2$$ $$H_0:\mu_1=\mu_2.\mu_1\neq\mu_2$$ (b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations. The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations. The standard normal. We assume that both population distributions are approximately normal with known standard deviations. The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations. (c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate. (Test the difference $$\mu_1 - \mu_2$$. Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.) (e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha? At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant. At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant. At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant. (f) Interpret your conclusion in the context of the application. Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver. Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver. Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for $$\mu_1 - \mu_2$$. (Round your answers to two decimal places.) lower limit upper limit (h) Explain the meaning of the confidence interval in the context of the problem. Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver. Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver. Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver. Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver. asked 2021-05-05 The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denotethe bulk density of Pima clay loam. Studies show that X is normally distributed with $$\displaystyle\mu={1.5}$$ and $$\displaystyle\sigma={0.2}\frac{{g}}{{c}}{m}^{{3}}$$. (a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability. (b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than $$\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}$$. (c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of $$\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}$$? Explain, based on theprobability of this occurring. (d) What point has the property that only 10% of the soil samples have bulk density this high orhigher? (e) What is the moment generating function for X? asked 2021-04-20 (1 pt) A new software company wants to start selling DVDs withtheir product. The manager notices that when the price for a DVD is19 dollars, the company sells 140 units per week. When the price is28 dollars, the number of DVDs sold decreases to 90 units per week.Answer the following questions: A. Assume that the demand curve is linear. Find the demand, q, as afunction of price, p. Answer: q= B. Write the revenue function, as a function of price. Answer:R(p)= C. Find the price that maximizes revenue. Hint: you may sketch thegraph of the revenue function. Round your answer to the closestdollar. Answer: D. Find the maximum revenue. Answer: asked 2021-01-04 A population of values has a normal distribution with $$\displaystyle\mu={198.8}$$ and $$\displaystyle\sigma={69.2}$$. You intend to draw a random sample of size $$\displaystyle{n}={147}$$. Find the probability that a sample of size $$\displaystyle{n}={147}$$ is randomly selected with a mean between 184 and 205.1. $$\displaystyle{P}{\left({184}{<}{M}{<}{205.1}\right)}=$$? Write your answers as numbers accurate to 4 decimal places. asked 2021-02-08 A population of values has a normal distribution with $$\displaystyle\mu={204.3}$$ and $$\displaystyle\sigma={43}$$. You intend to draw a random sample of size $$\displaystyle{n}={111}$$. Find the probability that a single randomly selected value is less than 191.2. $$\displaystyle{P}{\left({X}{<}{191.2}\right)}=$$? Find the probability that a sample of size $$\displaystyle{n}={111}$$ is randomly selected with a mean less than 191.2. $$\displaystyle{P}{\left({M}{<}{191.2}\right)}=$$? Write your answers as numbers accurate to 4 decimal places. asked 2021-03-05 A CI is desired for the true average stray-load loss A (watts) for a certain type of induction motor when the line current is heldat 10 amps for a speed of 1500 rpm. Assume that stray-load loss isnormally distributed with A = 3.0. In this problem part (a) wants you to compute a 95% CI for A when n =25 and the sample mean = 58.3. asked 2021-04-13 As depicted in the applet, Albertine finds herself in a very odd contraption. She sits in a reclining chair, in front of a large, compressed spring. The spring is compressed 5.00 m from its equilibrium position, and a glass sits 19.8m from her outstretched foot. a)Assuming that Albertine's mass is 60.0kg , what is $$\displaystyle\mu_{{k}}$$, the coefficient of kinetic friction between the chair and the waxed floor? Use $$\displaystyle{g}={9.80}\frac{{m}}{{s}^{{2}}}$$ for the magnitude of the acceleration due to gravity. Assume that the value of k found in Part A has three significant figures. Note that if you did not assume that k has three significant figures, it would be impossible to get three significant figures for $$\displaystyle\mu_{{k}}$$, since the length scale along the bottom of the applet does not allow you to measure distances to that accuracy with different values of k. asked 2021-02-23 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? asked 2020-12-30 Is the gift you purchased for that special someone really appreciated? This was the question investigated in the Journal of Experimental Social Psychology (Vol. 45, 2009). Toe researchers examined the link between engagement ring price (dollars) and level of appreciation of the recipient $$\displaystyle{\left(\text{measured on a 7-point scale where}\ {1}=\ \text{"not at all" and}\ {7}=\ \text{to a great extent"}\right)}.$$ Participants for the study were those who used a popular Web site for engaged couples. The Web site's directory was searched for those with "average" American names (e.g., "John Smith," "Sara Jones"). These individuals were then invited to participate in an online survey in exchange for a$10 gift certificate. Of the respondents, those who paid really high or really low prices for the ring were excluded, leaving a sample size of 33 respondents. a) Identify the experimental units for this study. b) What are the variables of interest? Are they quantitative or qualitative in nature? c) Describe the population of interest. d) Do you believe the sample of 33 respondents is representative of the population? Explain. e. In a second, designed study, the researchers investigated whether the link between gift price and level of appreciation was stronger for birthday gift givers than for birthday gift receivers. Toe participants were randomly assigned to play the role of gift-giver or gift-receiver. Assume that the sample consists of 50 individuals. Use a random number generator to randomly assign 25 individuals to play the gift-receiver role and 25 to play the gift-giver role.
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.