Hypothesis Testing - Worksheet PDF

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Hypothesis Testing Worksheet 1.

On average people in Wollongong commute 19.7 km to work. A manager of a large business in Wollongong wants to know whether their employees travel a below average distance. They sample 15 of their employees and find that they travel on average 14.2 km to get to work with a standard deviation of 7 km. Perform a hypothesis test for this employer with a significance level of 0.01.

2.

A soft drink company advertises that their soft drinks contain only 6 grams of sugar per bottle with a standard deviation of 1.5 grams. A parent thinks this claim is incorrect and that the bottles contain more sugar than advertised. They buy 4 bottles and find that the average sugar content per bottle is 7.8 grams. Perform a hypothesis test with a significance level of 0.05 to test the company’s claim.

3.

A researcher is investigating the amount of time watching television. They want to test if there is a difference in the average number of hours of television watched by 15-19 year olds compared to 20-24 year olds. They sample eight 15-19 year olds and find they average 19.5 hours of television per week with a standard deviation of 4.3 hours per week. They also sample eleven 20-24 year olds and find they average 16.2 hours of television per week with a standard deviation of 3.9 hours per week. Assuming the population standard deviations are equal, what will the researcher conclude at a significance level of 0.05?

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SOLUT IONS 1.

On average people in Wollongong commute 19.7 km to work. A manager of a large business in Wollongong wants to know whether their employees travel a below average distance. They sample 15 of their employees and find that they travel on average 14.2 km to get to work with a standard deviation of 7 km. Perform a hypothesis test for this employer with a significance level of 0.01. Relevant information: ฀฀0 = 19.7 km, “below”, ฀ ฀ = 15�, 1฀฀= 14.2 km, ฀฀1 = 7 km, ฀ ฀ = 0.01 Hypotheses: Left-tail so ฀฀0 : ฀฀1 ≥ 19.7 km and ฀฀1 : ฀฀1 < 19.7 km Type of test: ฀฀ is unknown so t-test Critical value: ฀฀฀฀ = 14 so ฀฀฀฀฀฀฀฀฀฀ = 2.6245 Decision rule: If ฀฀฀฀฀฀฀฀฀฀ < −2.6245 reject ฀฀0 in favour of ฀฀1 , otherwise do not reject ฀฀0 Test statistic: 14.2 − 19.7 ฀฀฀฀฀฀฀฀฀฀ = = −3.043 7⁄ √15 Conclusion: Reject ฀฀0 as −3.043 < −2.6245. We conclude that the employees travel a below average distance to work with a level of significance of 0.01.

2.

A soft drink company advertises that their soft drinks contain only 6 grams of sugar per bottle with a standard deviation of 1.5 grams. A parent thinks this claim is incorrect and that the bottles contain more sugar than advertised. They buy 4 bottles and find that the average sugar content per bottle is 7.8 grams. Perform a hypothesis test with a significance level of 0.05 to test the company’s claim. �1฀฀= 7.8 grams, ฀ ฀ = 0.05 Relevant information: ฀฀0 = 6 grams, ฀฀0 = 1.5 grams, “above”, ฀ ฀ = 4, Hypotheses: Right-tail so ฀฀0 : ฀฀1 ≤ 6 grams and ฀฀1 : ฀฀1 > 6 grams Type of test: Assume the stated population standard deviation so Z-test Critical value: ฀฀฀฀฀฀฀฀฀฀ = 1.6449 Decision rule: If ฀฀฀฀฀฀฀฀฀฀ > 1.6449 reject ฀฀0 in favour of ฀฀1 , otherwise do not reject ฀฀0 Test statistic: 7.8 − 6 ฀฀฀฀฀฀฀฀฀฀ = = 2.4 1.5⁄√4 Conclusion: Reject ฀฀0 as 2.4 > 1.64495. We conclude that the bottles contain more sugar than advertised with a level of significance of 0.05.

3.

A researcher is investigating the amount of time watching television. They want to test if there is a difference in the average number of hours of television watched by 15-19 year olds compared to 20-24 year olds. They sample eight 15-19 year olds and find they average 19.5 hours of television per week with a standard deviation of 4.3 hours per week. They also sample eleven 20-24 year olds and find they average 16.2 hours of television per week with a standard deviation of 3.9 hours per week. Assuming the population standard deviations are equal, what will the researcher conclude at a significance level of 0.05? Relevant information: “difference”, ฀฀1 = 8, �฀฀1 = 19.5 hours, ฀฀1 = 4.3 hours, ฀฀2 = 11, �฀฀ 2 = 16.2 hours, ฀฀2 = 3.9 hours, ฀ ฀ = 0.05 Hypotheses: Two-tail so ฀฀0 : ฀฀1 − ฀฀2 = 0 hours and ฀฀1 : ฀฀1 − ฀฀2 ≠ 0 hours Type of test: Assuming the same population standard deviation means pooled t-test Critical value: ฀฀฀฀ = 17 so ฀฀฀฀฀฀฀฀฀฀ = 2.1098 Decision rule: If ฀฀฀฀฀฀฀฀฀฀ < −2.1098 or ฀฀฀฀฀฀฀฀฀฀ > 2.1098 reject ฀฀0 in favour of ฀฀1 , otherwise do not reject ฀฀0 Test statistic: 7×4.32 +10×3.92

฀฀฀ ฀ = �

17

= 4.069 so ฀฀฀฀฀฀฀฀฀฀ =

19.5−16.2 1

1

4.07� + 11 8

= 1.745

Conclusion: Do not reject ฀฀0 as −2.1098 < 1.745 < 2.1098. We cannot conclude that there is a difference in hours of television watched between 15-19 year olds and 20-24 year olds at a level of significance of 0.01.

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