Question

The accompanying data represent the amount of soft drink filled in a sample of 50 consecutive 2-liter bottles. The results are listed horizontally in the order of being filled. Complete parts (a) through (e). a. At the 0.01 level of significance, is there evidence that the mean amount of soft drink filled is different from 2.0 liters? State the null and alternative hypotheses. Upper H0: μ = 2.0 Upper H1: μ ≠ 2.0 Identify the critical value(s). The critical value(s) is(are) nothing. (Round to four decimal places as needed. Use a comma to separate answers as needed.) Determine the test statistic. The test statistic is nothing. (Round to four decimal places as needed.) State the conclusion. Do not reject Upper H0. There is insufficient evidence to conclude that the mean amount of soft drink filled is different from 2.0 liters. b. Determine the p-value in (a) and interpret its meaning. The p-value is nothing. (Round to four decimal places as needed.) Interpret the meaning of the p-value. Choose the correct answer below. A. The p-value is the probability of not rejecting the null hypothesis when it is false. B. The p-value is the probability of obtaining a sample mean that is equal to or more extreme than 0.0007 liters above 2.0 liters if the null hypothesis is false. C. The p-value is the probability of obtaining a sample mean that is equal to or more extreme than 0.0007 liters below 2.0 liters if the null hypothesis is false. D. The p-value is the probability of obtaining a sample mean that is equal to or more extreme than 0.0007 liters away from 2.0 liters if the null hypothesis is true. c. In (a), you assumed that the distribution of the amount of soft drink filled was normally distributed. Evaluate this assumption by constructing a boxplot or a normal probability plot. Construct a normal probability plot. Choose the correct graph below. A. Normal probability plot Z-value Liters A normal probability plot has a horizontal axis labeled Z-value from negative 3 to 3 in increments of 1 and a vertical axis labeled Trough Width in inches from 1.85 to 2.15 in increments of 0.05. Plotted points begin at the point (negative 1.29, 1.94) and generally form a line that rises from left to right, ending at the point (1.29, 2.06). All coordinates are approximate. B. Normal probability plot Z-value Liters A normal probability plot has a horizontal axis labeled Z-value from negative 3 to 3 in increments of 1 and a vertical axis labeled Trough Width in inches from 1.85 to 2.15 in increments of 0.05. Plotted points begin at the point (negative 2.06, 1.97) and generally form a horizontal line, ending at the point (2.06, 1.97). All coordinates are approximate. C. Normal probability plot Z-value Liters A normal probability plot has a horizontal axis labeled Z-value from negative 3 to 3 in increments of 1 and a vertical axis labeled Trough Width in inches from 1.85 to 2.15 in increments of 0.05. Plotted points begin at the point (negative 2.06, 1.90) and generally form a line that rises from left to right, ending at the point (2.06, 2.11). All coordinates are approximate. d. Do you think that the assumption needed in order to conduct the t test in (a) is valid? A. No, because the normal probability plot is a straight line. The population distribution is not approximately uniform. B. No, because the normal probability plot is not a straight line. The population distribution is not approximately normal. C. Yes, because the normal probability plot is a straight line. The population distribution is approximately normal. D. Yes, because the normal probability plot is a straight line. The population distribution is approximately uniform. e. Examine the values of the 50 bottles in their sequential order, as given in the problem. Does there appear to be a pattern to the results? If so, what impact might this pattern have on the validity of the results in (a)? A. The amount of fill is increasing over time so the values are not independent. Therefore, the t test is valid. B. The amount of fill is increasing over time so the values are independent. Therefore, the t test is invalid. C. The amount of fill is decreasing over time so the values are not independent. Therefore, the t test is invalid. D. The amount of fill is decreasing over time so the values are independent. Therefore, the t test is valid. 

          The accompanying data represent the amount of soft drink filled in a sample of 50 consecutive 2-liter bottles. The results are listed horizontally in the order of being filled. Complete parts (a) through (e).

a. At the 0.01 level of significance, is there evidence that the mean amount of soft drink filled is different from 2.0 liters? State the null and alternative hypotheses.

Upper H0: μ = 2.0
Upper H1: μ ≠ 2.0

Identify the critical value(s). The critical value(s) is(are) nothing. (Round to four decimal places as needed. Use a comma to separate answers as needed.)

Determine the test statistic. The test statistic is nothing. (Round to four decimal places as needed.)

State the conclusion.
Do not reject Upper H0. There is insufficient evidence to conclude that the mean amount of soft drink filled is different from 2.0 liters.

b. Determine the p-value in (a) and interpret its meaning. The p-value is nothing. (Round to four decimal places as needed.)

Interpret the meaning of the p-value. Choose the correct answer below.
A. The p-value is the probability of not rejecting the null hypothesis when it is false.
B. The p-value is the probability of obtaining a sample mean that is equal to or more extreme than 0.0007 liters above 2.0 liters if the null hypothesis is false.
C. The p-value is the probability of obtaining a sample mean that is equal to or more extreme than 0.0007 liters below 2.0 liters if the null hypothesis is false.
D. The p-value is the probability of obtaining a sample mean that is equal to or more extreme than 0.0007 liters away from 2.0 liters if the null hypothesis is true.

c. In (a), you assumed that the distribution of the amount of soft drink filled was normally distributed. Evaluate this assumption by constructing a boxplot or a normal probability plot.

Construct a normal probability plot. Choose the correct graph below.
A. Normal probability plot
Z-value
Liters
A normal probability plot has a horizontal axis labeled Z-value from negative 3 to 3 in increments of 1 and a vertical axis labeled Trough Width in inches from 1.85 to 2.15 in increments of 0.05. Plotted points begin at the point (negative 1.29, 1.94) and generally form a line that rises from left to right, ending at the point (1.29, 2.06). All coordinates are approximate.

B. Normal probability plot
Z-value
Liters
A normal probability plot has a horizontal axis labeled Z-value from negative 3 to 3 in increments of 1 and a vertical axis labeled Trough Width in inches from 1.85 to 2.15 in increments of 0.05. Plotted points begin at the point (negative 2.06, 1.97) and generally form a horizontal line, ending at the point (2.06, 1.97). All coordinates are approximate.

C. Normal probability plot
Z-value
Liters
A normal probability plot has a horizontal axis labeled Z-value from negative 3 to 3 in increments of 1 and a vertical axis labeled Trough Width in inches from 1.85 to 2.15 in increments of 0.05. Plotted points begin at the point (negative 2.06, 1.90) and generally form a line that rises from left to right, ending at the point (2.06, 2.11). All coordinates are approximate.

d. Do you think that the assumption needed in order to conduct the t test in (a) is valid?
A. No, because the normal probability plot is a straight line. The population distribution is not approximately uniform.
B. No, because the normal probability plot is not a straight line. The population distribution is not approximately normal.
C. Yes, because the normal probability plot is a straight line. The population distribution is approximately normal.
D. Yes, because the normal probability plot is a straight line. The population distribution is approximately uniform.

e. Examine the values of the 50 bottles in their sequential order, as given in the problem. Does there appear to be a pattern to the results? If so, what impact might this pattern have on the validity of the results in (a)?
A. The amount of fill is increasing over time so the values are not independent. Therefore, the t test is valid.
B. The amount of fill is increasing over time so the values are independent. Therefore, the t test is invalid.
C. The amount of fill is decreasing over time so the values are not independent. Therefore, the t test is invalid.
D. The amount of fill is decreasing over time so the values are independent. Therefore, the t test is valid.
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Added by Scott W.

Elementary Statistics a Step by Step Approach
Elementary Statistics a Step by Step Approach
Allan G. Bluman 9th Edition
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The accompanying data represent the amount of soft drink filled in a sample of 50 consecutive 2-liter bottles. The results are listed horizontally in the order of being filled. Complete parts (a) through (e). a. At the 0.01 level of significance, is there evidence that the mean amount of soft drink filled is different from 2.0 liters? State the null and alternative hypotheses. Upper H0: μ = 2.0 Upper H1: μ ≠ 2.0 Identify the critical value(s). The critical value(s) is(are) nothing. (Round to four decimal places as needed. Use a comma to separate answers as needed.) Determine the test statistic. The test statistic is nothing. (Round to four decimal places as needed.) State the conclusion. Do not reject Upper H0. There is insufficient evidence to conclude that the mean amount of soft drink filled is different from 2.0 liters. b. Determine the p-value in (a) and interpret its meaning. The p-value is nothing. (Round to four decimal places as needed.) Interpret the meaning of the p-value. Choose the correct answer below. A. The p-value is the probability of not rejecting the null hypothesis when it is false. B. The p-value is the probability of obtaining a sample mean that is equal to or more extreme than 0.0007 liters above 2.0 liters if the null hypothesis is false. C. The p-value is the probability of obtaining a sample mean that is equal to or more extreme than 0.0007 liters below 2.0 liters if the null hypothesis is false. D. The p-value is the probability of obtaining a sample mean that is equal to or more extreme than 0.0007 liters away from 2.0 liters if the null hypothesis is true. c. In (a), you assumed that the distribution of the amount of soft drink filled was normally distributed. Evaluate this assumption by constructing a boxplot or a normal probability plot. Construct a normal probability plot. Choose the correct graph below. A. Normal probability plot Z-value Liters A normal probability plot has a horizontal axis labeled Z-value from negative 3 to 3 in increments of 1 and a vertical axis labeled Trough Width in inches from 1.85 to 2.15 in increments of 0.05. Plotted points begin at the point (negative 1.29, 1.94) and generally form a line that rises from left to right, ending at the point (1.29, 2.06). All coordinates are approximate. B. Normal probability plot Z-value Liters A normal probability plot has a horizontal axis labeled Z-value from negative 3 to 3 in increments of 1 and a vertical axis labeled Trough Width in inches from 1.85 to 2.15 in increments of 0.05. Plotted points begin at the point (negative 2.06, 1.97) and generally form a horizontal line, ending at the point (2.06, 1.97). All coordinates are approximate. C. Normal probability plot Z-value Liters A normal probability plot has a horizontal axis labeled Z-value from negative 3 to 3 in increments of 1 and a vertical axis labeled Trough Width in inches from 1.85 to 2.15 in increments of 0.05. Plotted points begin at the point (negative 2.06, 1.90) and generally form a line that rises from left to right, ending at the point (2.06, 2.11). All coordinates are approximate. d. Do you think that the assumption needed in order to conduct the t test in (a) is valid? A. No, because the normal probability plot is a straight line. The population distribution is not approximately uniform. B. No, because the normal probability plot is not a straight line. The population distribution is not approximately normal. C. Yes, because the normal probability plot is a straight line. The population distribution is approximately normal. D. Yes, because the normal probability plot is a straight line. The population distribution is approximately uniform. e. Examine the values of the 50 bottles in their sequential order, as given in the problem. Does there appear to be a pattern to the results? If so, what impact might this pattern have on the validity of the results in (a)? A. The amount of fill is increasing over time so the values are not independent. Therefore, the t test is valid. B. The amount of fill is increasing over time so the values are independent. Therefore, the t test is invalid. C. The amount of fill is decreasing over time so the values are not independent. Therefore, the t test is invalid. D. The amount of fill is decreasing over time so the values are independent. Therefore, the t test is valid.
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Transcript

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00:01 In this question, the given data is between x and px of x, that is the probability mass function of x between the given values.
00:10 Here, let us assume, x be the rated capacity of a freezer of a freezer of this brand sold at a certain store in hadry.
00:26 Now, computing the expected value of x, denoted by e of x, will be equal to x 1 times.
00:34 X1 plus x2 times p of x2 plus x3 times p of x3 which is equal to 16 times p of 16 plus 18 p of 18 plus 20 times p of 20 on substituting the values from the given data we get 16 times of 0 .2 plus 18 times of 0 .5 plus 20 times of 0 .5 plus 20 times of 0 .3.
01:05 And calculating this, we get the expected value e of x is equal to 18 .2.
01:11 Therefore, the expected capacity of a freezer of a brand is 18 .2.
01:18 Now next, computing the value of e of x squared is equal to x1 square of p of x2 plus x2 square times of p of x2 plus x3 square times p of x3 and so on till xn square times p of xin.
01:39 On computing the values were, we get 16 square times p of 16 plus 18 square times p of 18 plus 20 square times p of 20.
01:51 On substituting the values here, we get the final solution of e of x squared.
01:57 Be equal to 33 .2.
02:01 Therefore, the expected value of x square is 33 .3 .2.
02:09 Now, the variance can be computed as variance denoted by v of x is equal to e of x squared minus e of x whole square.
02:21 This is equal to 333 .2 minus 18 .2 both square...
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