For Some Positive Value Of Z The Probability That A Standard
For some positive value of Z the probability that a standard normal Va
Cleaned Assignment Instructions
For some positive value of Z, the probability that a standard normal variable is between 0 and Z is 0.3770. The value of Z is to be determined among the options: 0.18, 0.81, 1.16, 1.47.
True or False: A worker earns $15 per hour at a plant and is told that only 2.5% of all workers make a higher wage. If the wage is assumed to be normally distributed and the standard deviation of wage rates is $5 per hour, the average wage for the plant is $7.50 per hour.
The owner of a fish market has an assistant who has determined that the weights of catfish are normally distributed, with mean of 3.2 pounds and standard deviation of 0.8 pound. What percentage of samples of 4 fish will have sample means between 3.0 and 4.0 pounds?
True or False: If the amount of gasoline purchased per car at a large service station has a population mean of $15 and a population standard deviation of $4 and a random sample of 4 cars is selected, there is approximately a 68.26% chance that the sample mean will be between $13 and $17.
The t distribution: assumes the population is normally distributed; approaches the normal distribution as the sample size increases; has more area in the tails than does the normal distribution; all of the above.
True or False: A sample of 100 fuses from a very large shipment is found to have 10 that are defective. The 95% confidence interval would indicate that, for this shipment, the proportion of defective fuses is between 0 and 0.28.
Suppose we want to test H₀: μ ≥ 30 versus H₁: μ
True or False: The larger is the p-value, the more likely one is to reject the null hypothesis.
What do we mean when we say that a simple linear regression model is ‘statistically’ useful?
Which of the following statements about the method of exponential smoothing is not true?
A company that manufactures designer jeans is contemplating whether to increase its advertising budget by $1 million for next year. If the expanded advertising campaign is successful, sales increase by $1.6 million; if it fails, by $400,000; if no increase, by $200,000. Identify the outcomes in this decision-making problem.
The data below represents the grams of carbohydrates in a sample servings of breakfast cereal.
The coefficient of variation for this data would be answered as a percentage, consistent with examples in Levine et al.'s text, between one or two decimal places, e.g., 12.3 or 12.34.
In a local cellular phone area: Company A has 70% market share, with 2% interference rate; Company B has 30% share, with 3% interference rate. Given a call with interference, what is the probability it came from Company B?
The manager of a service station believes that motorists change their oil less frequently than the recommended twice per year. In a survey of 15 car owners, what is the approximate value of the test statistic?
What decision would be made if a hypothesis test on this problem has H₀: μ ≥ 2 and H₁: μ
Using a dataset of sales for a department store over ten months, what is the MAPE using a three-period moving average?
Using simple exponential smoothing with a smoothing constant of 0.2, what is the MAPE for this forecast model?
From a sample of 25 batches of 500 chips, defects range from 1 to 29. If constructing a frequency distribution starting with '0 but less than 5', what is the relative frequency of the '10 but less than 15' class?
Power tool manufacturer claims a mean assembly time of 50 minutes with a standard deviation of 40 minutes; in a sample of 64, what is the probability the mean is less than 46 minutes?
A manufacturer measures the length of insulation sheets. The standard deviation is 0.15 metres; with a sample of 144, what are the 90% confidence limits for the mean length?
Regarding an insurance company's data: what is the least squares estimate of the slope between income and insurance amount? What is the intercept? And, for a family earning $85,000, what is the predicted insurance? For a family earning $90,000, what is the residual?
International Pictures considers releasing a movie 'Claws'. Boundaries include potential success levels, profits, and associated probabilities; analysis involves opportunity loss, optimal action, and valuation of forecast accuracy. Also included are probabilities concerning preview ratings, success levels, and revenue forecasts, with a focus on data analysis methods like moving averages, exponential smoothing, and regression for seasonal transit data and marketing investment decisions.
Sample Paper For Above instruction
The problem involves a series of statistical and probabilistic applications, requiring mastery over concepts such as standard normal distribution, hypothesis testing, confidence intervals, regression analysis, and forecasting methods. We explore each scenario thoroughly, integrating theory with calculation to interpret real-world data and decision-making implications.
Analysis of Standard Normal Probability and Z-Score Calculation
The first question asks for the Z-value corresponding to a probability of 0.3770 between 0 and Z in the standard normal distribution. Using the symmetry of the normal curve and the z-score table, we identify that a cumulative area of 0.3770 beyond 0 corresponds roughly to Z ≈ 1.16. This is derived from the fact that the area from 0 to Z is 0.3770; thus, the cumulative area from the far left to Z is 0.5 + 0.3770 = 0.8770, leading to Z ≈ 1.16 (from standard normal tables). Therefore, the correct choice among the options provided is 1.16, which closely corresponds to the calculated value and demonstrates a fundamental understanding of normal probability.
Mean Wage Estimation from Normal Distribution
The second question assesses the understanding of normal distribution parameters in the context of wages. Given that only 2.5% of workers earn more than a certain wage, this corresponds to the top 2.5% tail on the normal curve, associated with a Z-score of approximately 1.96 (using standard Z-tables). Using the formula:
Wage = μ + Z σ, we solve for μ: μ = W - Z σ. Substituting the known values, W = 15, Z ≈ 1.96, σ = 5, we check whether μ equals $7.50. Actual calculations show that for Z = 1.96, the cutoff wage W would be μ + 1.96 * 5. From the given data, the claim that the mean wage is $7.50 is false, because with the standard normal distribution, the mean would be substantially higher, confirming the false statement—highlighting the importance of understanding standard scores and their relation to distribution parameters.
Sample Means and Normal Approximation
The third question involves sample means distribution, where weights of catfish are normally distributed with a mean of 3.2 pounds, standard deviation 0.8, and sample size 4. The sampling distribution of the mean for samples of size 4 has a standard error of σ/√n = 0.8/2 = 0.4. To find the percentage of samples with means between 3.0 and 4.0 pounds, we convert the bounds into Z-scores:
Z for 3.0: (3.0 - 3.2)/0.4 = -0.5
Z for 4.0: (4.0 - 3.2)/0.4 = 2.0
The probability between these Z-scores is the sum of the areas between -0.5 and 2.0, which from standard normal tables is approximately 0.6922 or 69.2%, close to the 67% option, indicating that approximately 67% of such sample means will fall within this range.
Sampling Distribution of the Mean with Known Population Parameters
In the fourth problem, with a population mean of $15, standard deviation of $4, and sample size 4, the standard error is 4/√4 = 2. The probability that the sample mean is between $13 and $17 is calculated as:
Z for 13: (13 - 15)/2 = -1.0
Z for 17: (17 - 15)/2 = 1.0
Using standard normal distribution, the probability between Z = -1 and 1 is approximately 68.26%. This aligns with the empirical rule, confirming the statement’s correctness and underlining the importance of standard error in inferential statistics.