NAME: _______________________
Date:________________________
REVIEW PROBLEMS SET-3
QUESTIONS 1-8 The Interstate Conference of Employment Security Agencies is conducting a study to determine the number of hours per week worked in the United States. A mean of 36 hours per week was obtained from a random sample of 20 workers and the standard deviation of the sample was 8 hours. Assume hours worked per week are normally distributed in the population. Q1. The point estimate of the mean number of hours worked by the population US workers is: _______ Q2. The margin of error for a 99% confidence interval for the mean number of hours worked per week for the population of US workers is _________
Q3. The 90% confidence interval for the mean number of hours worked per week for the population of US workers is _________________ Suppose the agency believes the average workweek in the United States has decreased from the traditional 40hrs per week largely because of a rise in part-time workers and they will use this sample on to conduct a hypothesis test.
Q4. Which of the following pairs of hypotheses will they need to test ? Why a) Ho : µ >=36 ; Ha : µ < 36 b) Ho : µ = 36 ; Ha : µ ≠ 36 c) Ho : µ >=40 ; Ha : µ < 40 d) Ho : µ = 40 ; Ha : µ ≠ 40 e) None of the above
Q5. What is the p-value for their test ?
Q6 What is the value of the test statistic ?
Q7. Using a significance level alpha=5%, the correct conclusion is : a) Reject Ho, there is no significant evidence that the average number of hours per week has decreased. b) Reject Ho, there is evidence that the average number of hours per week has decreased. c) Do not Reject Ho, there is evidence that the average number of hours per week has decreased. d) Do not Reject Ho, there is no evidence that the average number of hours per week has decreased.
Q8. Using a significance level alpha=1%, the correct conclusion is : a) Reject Ho, there is no significant evidence that the average number of hours per week has decreased. b) Reject Ho, there is evidence that the average number of hours per week has decreased. c) Do not Reject Ho, there is evidence that the average number of hours per week has decreased. d) Do not Reject Ho, there is no evidence that the average number of hours per week has decreased.
QUESTIONS 9 Compute a 95% confidence interval for the population proportion for a sample size = 200 and a sample proportion = 25%
QUESTION 10 What is the margin of error of a 99% confidence interval for the population mean based on the following info: X= 15, σ=4, n=10.? QUESTION 11. What is the margin of error of a 99% confidence interval for the population mean based on the following info: X= 15, s=4, n=10.?
NAME: _______________________
Date:________________________ QUESTIONS 12-14 A company wishes to determine if their training program is helping workers perform better on their industrial plant. Four workers are selected at random to undergo the training. The four workers take a performance test prior to the training and another one after the training. The scores are given below.
Worker’s score 1 2 3 4
First score 450 520 720 600
Second score 440 600 720 630
Assume that the change in score (second score – first score) for the population of all workers is normally distributed with mean μ. Q12. The margin of error of a 99% confidence interval for μ is (Use 3 decimals) Q13. What is the p-value for testing the hypothesis that the training has been effective in improving the performance of workers ? Q14. Is there significant evidence at the 1% level that the training has been effective in improving the performance of workers ? QUESTIONS 15 A store manager wants to use a 95% confidence interval to estimate the average total monthly purchases per customer at the store within $20 of the actual average. Historical data shows that the standard deviation of total monthly sales for all customers is about $40. How large a sample do they need ? QUESTION 16 A study will be conducted to construct a 90% confidence interval for a population proportion. An error of 0.03 is desired. There is no knowledge as to what the population proportion will be. What sample size is required ? QUESTIONS 17-20 A plant manufactures digital components that are supposed to be 15mm in length. The quality control department periodically checks the process to determine if length of components produced is in accordance with specifications. If the components do not comply with the required length, the process needs adjustment. Suppose five components are randomly selected and their average length is 13mm with a standard deviation of 4mm. Q17. What would be the conclusion using a significant level alpha= 5%.
a) Reject Ho, the process needs adjustment. b) Reject Ho, the process DOES NOT need adjustment. c) Do not Reject Ho, the process needs adjustment. d) Do not Reject Ho, the process DOES NOT need adjustment.
Q18. What is the p-value of the hypothesis test ? Q19. What is the value of the test statistic ? Q20. Will the conclusion be the same using a significance level alpha =1% ?
INDEX
Templates for: | INFERENCE TEMPLATES | |
Prepared By : | Dr. Gladys E. Simpson | |
Last Revision: | Nov 2018 | |
INDEX OF TEMPLATES. | ||
Topic / Problem | SINGLE POPULATION | TWO POPULATIONS |
Z Conf. Interval -MEAN-Pop Std Deviation σ KNOWN | Z Conf. Interval -TWO MEANS-Pop Std Dev σ KNOWN | |
CONFIDENCE INTERVALS (CI) | T Conf. Interval -MEAN-Pop Std Dev σ UNKNOWN | T Conf. Interval -TWO MEANS-Pop Std Dev σ UNKNOWN |
margin of error | Z Conf. Interval-ONE PROPORTION | Z Conf. Interval -TWO PROPORTIONS |
HYPOTHESIS TESTING | Z TEST-ONE MEAN- Pop Std Dev σ KNOWN | Z TEST-TWO MEANS-σ KNOWN |
P-Values, test statistic | T TEST-ONE MEAN- Pop Std Dev σ UNKNOWN | T TEST-TWO MEANS-σ UNKNOWN |
Z TEST-ONE PROPORTION | Z TEST - TWO PROPORTIONS | |
SAMPLE SIZE N | SAMPLE SIZE For PROPORTIONS | |
SAMPLE SIZE FOR MEANS |
SAMPLE SIZE For MEANS
FINDING THE SAMPLE SIZE ( n ) for Estimation of MEANS | Back to index Page | |||||
Fill in ORANGE spaces | ||||||
What is the confidence level ? | ||||||
hence, alpha = | 100% | |||||
critical z* = | - 0 | |||||
What is the desired error of estimation ? | ||||||
What is the Standard deviation ? | ||||||
n | ERROR:#DIV/0! | |||||
Sample size required : | ERROR:#DIV/0! | |||||
SAMPLE SIZE For PROPORTIONS
FINDING THE SAMPLE SIZE ( n ) for PROPORTIONS | Back to index Page | |||||
Fill in ORANGE spaces | ||||||
What is the confidence level ? | ||||||
hence, alpha = | 100% | |||||
critical z* = | - 0 | |||||
What is the desired error of estimation ? | ||||||
Approixmation of p (use 0.5 if unknown) | ||||||
n | ERROR:#DIV/0! | |||||
Sample size required : | ERROR:#DIV/0! | |||||
CI -MEAN-σ KNOWN - using Z
FINDING A CONFIDENCE INTERVAL For a POPULATION MEAN | Back to index Page | ||||||
USING EXCEL TO FIND A CONFIDENCE INTERVAL FOR THE MEAN OF A POPULATION | Z Confidence interval | ||||||
Fill in ORANGE spaces | σ KNOWN | ||||||
What is the confidence level ? | |||||||
hence, alpha = | 100% | ||||||
critical Z* | 0.000 | ||||||
What is the sample mean ? (Point Estimate) | Finite Population ? | No | |||||
What is the sample size (n) ? | 1500 | ||||||
What is the POPULATION Standard deviation (σ ) ? | 1.000 | ||||||
Standard Error | ERROR:#DIV/0! | ||||||
Margin of Error | ERROR:#DIV/0! | ||||||
0% Confidence Interval | 0 | + / - | ERROR:#DIV/0! | ||||
From | ERROR:#DIV/0! | to | ERROR:#DIV/0! | ||||
lower limit | upper limit | ||||||
CI -MEAN-σ UNKNOWN - using t
FINDING A CONFIDENCE INTERVAL For a Population MEAN | Back to index Page | |||||
USING EXCEL TO FIND A CONFIDENCE INTERVAL FOR THE MEAN OF A POPULATION | T Confidence Interval | |||||
Fill in ORANGE spaces | σ UNKNOWN | |||||
What is the confidence level ? | 99% | |||||
hence, alpha = | 1% | |||||
Degrees of Freedom | 4 | |||||
Critical t* for 4 df | 4.604 | |||||
What is the sample mean ? (Point Estimate) | ||||||
What is the sample size (n) ? | 5 | |||||
What is the SAMPLE Standard deviation ? | ||||||
Standard Error : | - 0 | |||||
Margin of Error | - 0 | |||||
99% Confidence Interval | 0.000 | ± | - 0 | |||
From | 0.000 | to | - 0 | |||
lower limit | upper limit | |||||
CI-ONE PROPORTION
FINDING A CONFIDENCE INTERVAL For a PROPORTION | Back to index Page | ||||||||
Fill in ORANGE spaces | |||||||||
What is the confidence level ? | |||||||||
hence, alpha = | 100% | ||||||||
critical Z* | - 0 | ||||||||
X - Numbero of successes | ( if not given, use the proportion given to compute it: x = p * n ) | ||||||||
What is the sample size (n) ? | |||||||||
Sample Proportion p (Point Estimate) | ERROR:#DIV/0! | = X / n ( enter it directly if necessary ) | |||||||
Std Error | ERROR:#DIV/0! | sqrt( p * ( 1 - p ) / n ) | |||||||
Margin of Error | ERROR:#DIV/0! | ( critical z * std Error ) | |||||||
0% Confidence Interval | ERROR:#DIV/0! | ± | ERROR:#DIV/0! | or | From | ERROR:#DIV/0! | to | ERROR:#DIV/0! | in % form |
ERROR:#DIV/0! | ± | ERROR:#DIV/0! | or | From | ERROR:#DIV/0! | to | ERROR:#DIV/0! | in decimal form | |
lower limit | upper limit | ||||||||
CI -TWO MEANS-σ KNOWN
FINDING A CONFIDENCE INTERVAL For COMPARING TWO MEANS | Back to index Page | ||||
Fill in ORANGE spaces | Z Confidence interval | ||||
What is the confidence level ? | σ KNOWN | ||||
hence, alpha = | 100% | ||||
critical Z* | 0.000 | ||||
Sample 1 | Sample 2 | ||||
What is the sample mean ? | |||||
What is the sample size (n) ? | |||||
What is the Population Std deviation (σ )? | |||||
Mean Difference (x1 - x2 ) | 0.000 | ||||
Sampling Distrib Std dev (Std Error) | ERROR:#DIV/0! | ||||
Margin of Error | ERROR:#DIV/0! | ||||
0% Confidence Interval | 0.000 | ± | ERROR:#DIV/0! | ||
From | ERROR:#DIV/0! | to | ERROR:#DIV/0! | ||
lower limit | upper limit | ||||
CI -TWO MEANS-σ UNKNOWN
FINDING A CONFIDENCE INTERVAL For COMPARING TWO MEANS | Back to index Page | ||||
Fill in ORANGE spaces | T Confidence Interval | ||||
What is the confidence level ? | σ UNKNOWN | ||||
hence, alpha = | 100% | ||||
critical t* | ERROR:#NUM! | ||||
Sample 1 | Sample 2 | ||||
What is the sample mean ? | |||||
What is the sample size (n) ? | |||||
What is the Sample Standard deviation (s) ? | |||||
Degrees of Freedom | -1 | -1 | |||
Mean Difference (x1 - x2 ) | 0.000 | ||||
Standard Error : | ERROR:#DIV/0! | Assuming Unknown and Equal Pop variances | |||
Margin of Error | ERROR:#NUM! | ||||
0% Confidence Interval | 0.000 | ± | ERROR:#NUM! | ||
From | ERROR:#NUM! | to | ERROR:#NUM! | ||
lower limit | upper limit | ||||
CI-TWO PROPORTIONS
CONFIDENCE INTERVAL For COMPARING TWO PROPORTIONS | Back to index Page | |||||||
Fill in ORANGE spaces | ||||||||
What is the confidence level ? | ||||||||
hence, alpha = | 100% | |||||||
critical z* | - 0 | |||||||
Sample 1 | Sample 2 | |||||||
X - Numbero of successes | ||||||||
What is the sample size (n) ? | ||||||||
Difference (D) | ||||||||
What is the sample Proportion p | ERROR:#DIV/0! | ERROR:#DIV/0! | ERROR:#DIV/0! | ERROR:#DIV/0! | ||||
Std Error | ERROR:#DIV/0! | ERROR:#DIV/0! | ERROR:#DIV/0! | |||||
Margin of Error | ERROR:#DIV/0! | ( critical z * std Error ) | P-Value for D=0 | ERROR:#DIV/0! | ||||
0% Confidence Interval | ERROR:#DIV/0! | ± | ERROR:#DIV/0! | --> From | ERROR:#DIV/0! | to | ERROR:#DIV/0! | in % form |
ERROR:#DIV/0! | ± | ERROR:#DIV/0! | --> From | ERROR:#DIV/0! | to | ERROR:#DIV/0! | in decimal form | |
Z TEST-ONE MEAN-σ KNOWN
SIGNIFICANCE TEST - Hypothesis test For a SINGLE Population Mean When σ KNOWN | Back to index Page | ||||||
Fill in ORANGE spaces | ONE SAMPLE Z-TEST | ||||||
σ is the POPULATION STANDARD DEVIATION | |||||||
Ho: Pop mean( µ ) = | Significance level | When σ is KNOWN we use the Z tables, the Normal Distribution | |||||
TABLE A on book, or NORMDIST Excel Function | |||||||
Sample mean (x) | Finite Population ? | No | |||||
Sample size ( n ) | |||||||
Population std dev ( σ ) = | 1.000 | ||||||
smpl distr std dev | ERROR:#DIV/0! | ||||||
test statistic z | ERROR:#DIV/0! | ||||||
ONE SIDED ALTERNATIVE | TWO SIDED | ||||||
Hypotheses | Ho : µ >= | 0 | Ho : µ <= | 0 | Ho : µ = | 0 | |
Ha : µ < | 0 | Ha : µ > | 0 | Ha : µ ≠ | 0 | ||
P Values | P-value | ERROR:#DIV/0! | P-value | ERROR:#DIV/0! | P-value | ERROR:#DIV/0! | |
T TEST-ONE MEAN-σ UNKNOWN
HYPOTHESIS TEST About A POPULATION MEAN When σ UNKNOWN | Back to index Page | |||||
Fill in ORANGE spaces | ONE SAMPLE T-TEST | |||||
When σ ( the POPULATION STANDARD DEVIATION) is NOT KNOWN | ||||||
Ho: Pop mean( µ ) = | Significance level | We use the Sample Standard deviation to estimate it | ||||
and we work with a T DISTRIBUTION for the test statistic t | ||||||
Sample mean (x) | Using TABLE D on the BOOK or TDIST Function in Excel | |||||
Sample size ( n ) | ||||||
Sample std dev ( s ) = | ||||||
degrees of freedom | ||||||
Standard Error | ERROR:#DIV/0! | |||||
test statistic t | ERROR:#DIV/0! | |||||
1 SIDED ALTERNATIVE | TWO SIDED | |||||
Hypotheses | Ho : µ >= | 0 | Ho : µ <= | 0 | Ho : µ = | 0 |
Ha : µ < | 0 | Ha : µ > | 0 | Ha : µ ≠ | 0 | |
P Values | P-value | ERROR:#DIV/0! | P-value | ERROR:#DIV/0! | P-value | ERROR:#DIV/0! |
Z TEST-PROPORTION
HYPOTHESIS TEST About A POPULATION PROPORTION | Back to index Page | Prepared by Gladys Simpson [email protected] | |||||||||
Fill in ORANGE spaces | Z-TEST FOR A PROPORTION | If the sample size is too small, inference for proportions must be based on binomial distribution | |||||||||
when the sample is large, both the count X and the sample proportion are | |||||||||||
Ho: Pop proportion p0 = | Significance level (alpha) | approximately normal and we work with the normal distribution. | |||||||||
Here we work with the normal distribution. | |||||||||||
X number of successes | |||||||||||
IT is recommended to use large-sample z significance tst as long as the | |||||||||||
Sample size ( n ) | expected number of successes ( n*p) and expected failures (n*q) are both greater than 5 | ||||||||||
Sample Proportion p | ERROR:#DIV/0! | (x/n) - Number of successes / sample size | CAN'T USE THIS TEST when sample is too small | ||||||||
Hence, pop std dev = | - 0 | ||||||||||
Standard Error | ERROR:#DIV/0! | ||||||||||
test statistic z | ERROR:#DIV/0! | ||||||||||
1 SIDED ALTERNATIVE | TWO SIDED | ||||||||||
Hypotheses | Ho : p >= | 0% | Ho : p < = | 0% | Ho : p = | 0% | |||||
Ha : p < | 0% | Ha : p > | 0% | Ha : p ≠ | 0% | ||||||
P Values | P-value | ERROR:#DIV/0! | P-value | ERROR:#DIV/0! | P-value | ERROR:#DIV/0! | |||||
Z TEST-TWO MEANS-σ KNOWN
COMPARING TWO MEANS - Hypothesis test for Differences in sample means - σ KNOWN | Back to index Page | ||||||||
Fill in ORANGE spaces | TWO SAMPLE Z-TEST | When σ ( the POPULATION STANDARD DEVIATION) is KNOWN | |||||||
Significance level | |||||||||
Sample 1 | Sample 2 | We use the Sample Standard deviation to estimate it | |||||||
Sample mean (x) | we work with a Z DISTRIBUTION for the test statistic (Z) | ||||||||
Sample size ( n ) | |||||||||
Population std dev ( σ ) = | |||||||||
Mean difference (x1-x2) | 0.00 | ||||||||
Standard Error | ERROR:#DIV/0! | ||||||||
test statistic Z | ERROR:#DIV/0! | ||||||||
1 SIDED ALTERNATIVE | TWO SIDED | ||||||||
Hypotheses | Ho : µ1 = µ2 | Ho : µ1 = µ2 | Ho : µ1 = µ2 | ||||||
Ha : µ1 < µ2 | Ha : µ1 > µ2 | Ha : µ1 ≠ µ2 | |||||||
P Values | P-value | ERROR:#DIV/0! | P-value | ERROR:#DIV/0! | P-value | ERROR:#DIV/0! | |||
T TEST-TWO MEANS-σ UNKNOWN
COMPARING TWO MEANS - Hypothesis test for Difference in TWO POPULATION MEANS | Back to index Page | ||||||||
Fill in ORANGE spaces | TWO SAMPLE T-TEST | When σ ( the POPULATION STANDARD DEVIATION) is NOT KNOWN | |||||||
Significance level | We use the Sample Standard deviation to estimate it | ||||||||
and we work with a T DISTRIBUTION for the test statistic t | |||||||||
Sample 1 | Sample 2 | Using TABLE D on the BOOK or TDIST Function in Excel | |||||||
Sample mean (x) | |||||||||
Sample size ( n ) | |||||||||
Sample std dev ( s ) = | |||||||||
Mean difference (x1-x2) | 0.000 | ||||||||
degrees of freedom | -1 | -1 | |||||||
Standard Error | ERROR:#DIV/0! | Assuming Equal Variances | |||||||
test statistic t | ERROR:#DIV/0! | ||||||||
1 SIDED ALTERNATIVE | TWO SIDED | ||||||||
Hypotheses | Ho : µ1 = µ2 | Ho : µ1 = µ2 | Ho : µ1 = µ2 | ||||||
Ha : µ1 < µ2 | Ha : µ1 > µ2 | Ha : µ1 ≠ µ2 | |||||||
P Values | P-value | ERROR:#DIV/0! | P-value | ERROR:#DIV/0! | P-value | ERROR:#DIV/0! | |||
Z TEST - TWO PROPORTIONS
FINDING THE P-VALUE and HYPOTHESIS TEST for PROPORTIONS | Back to index Page | Prepared by Gladys Simpson [email protected] | |||||||||
Fill in ORANGE spaces | Z-TEST FOR COMPARING TWO PROPORTIONS | ||||||||||
Significance level | |||||||||||
Sample 1 | Sample 2 | Overall | |||||||||
X number of successes | 0 | ||||||||||
Sample size ( n ) | 0 | ||||||||||
Sample Proportion p | ERROR:#DIV/0! | ERROR:#DIV/0! | ERROR:#DIV/0! | ||||||||
Standard Error | ERROR:#DIV/0! | ERROR:#DIV/0! | ERROR:#DIV/0! | ||||||||
test statistic Z | ERROR:#DIV/0! | ||||||||||
Hypotheses | Ho : p1 = p2 | Ho : p1 = p2 | Ho : p1 = p2 | ||||||||
Ha : p1 < p2 | Ha : p1 > p2 | Ha : p1 ≠ p2 | |||||||||
P Values | P-value | ERROR:#DIV/0! | P-value | ERROR:#DIV/0! | P-value | ERROR:#DIV/0! | |||||

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