# Regarding z-scores: a. How is a z-score obtained? b. What is the interpretation of a z-score? c. An observation has a z-score of 2.9. Roughly speaking, what is the relative standing of the observation?

Question
Modeling data distributions
Regarding z-scores: a. How is a z-score obtained? b. What is the interpretation of a z-score? c. An observation has a z-score of 2.9. Roughly speaking, what is the relative standing of the observation?

2020-12-30
a. Z-score: The Z-score measures a particular value’s relationship to the mean in a group of values. It gives the number of standard deviation from the mean for the particular data point. The general formula to obtain Z-score is, $$Z=\frac{\text{x – population mean}}{\text{population standard deviation}}$$ b. Consider the following cases to interpret Z-scores: Case1: If a Z-score of −3 is obtained for a data point, then it can be interpreted that the data point is 3 standard deviations below the mean. Case2: If a Z-score of 0 is obtained for a data point, then it can be interpreted that the data point is 0 standard deviations from the mean. That is, the data point is equal to the mean. Case3: If a Z-score of 3 is obtained for a data point, then it can be interpreted that the data point is 3 standard deviations above the mean. c. The observation is very high because for normal distributions, about 95% of values are within two standard deviations.

### Relevant Questions

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
$$\mu_1 - \mu_2$$.
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.
The following table represents the Frequency Distribution and Cumulative Distributions for this data set: 12, 13, 17, 18, 18, 24, 26, 27, 27, 30, 30, 35, 37, 41, 42, 43, 44, 46, 53, 58 Class Frequency Relative Cumulative Frequency Frequency 10 but less than 20 5 20 but less than 30 4 30 but less than 4 4 40 but less than 50 5 50 but less than 60 2 TOTAL What is the Relative Frequency for the class: 50 but less than 60? State you answer as a value with exactly two digits after the decimal. for example 0.30 or 0.35
The following table represents the Frequency Distribution and Cumulative Distributions for this data set: 12, 13, 17, 18, 18, 24, 26, 27, 27, 30, 30, 35, 37, 41, 42, 43, 44, 46, 53, 58 Class Frequency Relative Cumulative Frequency Frequency 10 but less than 20 5 20 but less than 30 4 30 but less than 4 4 40 but less than 50 5 50 but less than 60 2 TOTAL What is the Relative Frequency for the class: 20 but less than 30? State you answer as a value with exactly two digits after the decimal. for example 0.30 or 0.35
The following table represents the Frequency Distribution and Cumulative Distributions for this data set: 12, 13, 17, 18, 18, 24, 26, 27, 27, 30, 30, 35, 37, 41, 42, 43, 44, 46, 53, 58 Class Frequency Relative Cumulative Frequency Frequency 10 but less than 20 5 20 but less than 30 4 30 but less than 4 4 40 but less than 50 5 50 but less than 60 2 TOTAL What is the Relative Frequency for the class: 40 but less than 50? State you answer as a value with exactly two digits after the decimal. for example 0.30 or 0.35
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1. The standard error of the estimate is the same at all points along the regression line because we assumed that A. The observed values of y are normally distributed around each estimated value of y-hat. B. The variance of the distributions around each possible value of y-hat is the same. C. All available data were taken into account when the regression line was calculated. D. The regression line minimized the sum of the squared errors. E. None of the above.
Look Out! A snowball rolls off a barn roof that slopes downward at an angle of 40 degrees . The edge of the roof is 14.0 m above the ground, and the snowball has a speed of 7.00 m/s as it rolls off the roof. Ignore air resistance.
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