Determining Whether a Z-distribution or T-distribution is Appropriate

  • 1.

    A designer wants to use sequins on her next dress that are more reflective than the majority of all other sequins. She collects a random sample of sequins and measures their reflectivity, finding a sample mean of 36% with a standard deviation of 1.2%. If the designer was to make a confidence interval or conduct a hypothesis test, should a z-distribution or a t-distribution be used?

    Answers:

    • A z-distribution should be used because the sample size is sufficiently large

    • A t-distribution should be used because the population standard deviation is not known

    • Neither table should be used because the data are not known to be normally distributed

    • A t-distribution should be used because the sample size is too small

  • 2.

    When building a fence, you buy 100 planks of wood, and measure the widths of five of them, calculating a mean of 0.77 inches. The manufacturer of these planks states their width's standard deviation is 0.005 inches. If you were to make a confidence interval or conduct a hypothesis test, should a z-distribution or a t-distribution be used?

    Answers:

    • Neither table should be used because the data are not known to be normally distributed

    • A t-distribution should be used because the sample size is too small

    • A t-distribution should be used because the population standard deviation is not known

    • A z-distribution should be used because the sample size is sufficiently large

  • 3.

    A dog walker is trying to get their business started in a new city. They survey 40 dog owners inquiring for how long they each walk their dogs, coming up with a sample mean of 35 minutes with a standard deviation of 4 minutes. If they were to make a confidence interval or conduct a hypothesis test, should a z-distribution or a t-distribution be used?

    Answers:

    • A t-distribution should be used because the population standard deviation is not known

    • A t-distribution should be used because the sample size is too small

    • A z-distribution should be used because the sample size is sufficiently large

    • Neither table should be used because the data are not known to be normally distributed

  • 4.

    Out of curiosity, a cat groomer takes a sample of how much fur is left by each cat he grooms, by weight. The groomer weighs the brushed fur of 83 cats throughout the week, and calculates a mean of 14.3 grams, with a standard deviation of 3.2 grams. If the groomer was to make a confidence interval or conduct a hypothesis test, should a z-distribution or a t-distribution be used?

    Answers:

    • Neither table should be used because the data are not known to be normally distributed

    • A z-distribution should be used because the sample size is sufficiently large

    • A t-distribution should be used because the sample size is too small

    • A t-distribution should be used because the population standard deviation is not known

  • 5.

    A flower arranger working with weddings and singular vase arrangements tabulates her year-end transactions to determine her mean profit per client. She calculates a mean of $302.34 with a standard deviation of $93.32. She hopes to compare this to other florists in her area. If she was to make a confidence interval or conduct a hypothesis test, should a z-distribution or a t-distribution be used?

    Answers:

    • A t-distribution should be used because the sample size is too small

    • A t-distribution should be used because the population standard deviation is not known

    • A z-distribution should be used because the sample size is sufficiently large

    • Neither table should be used because the data are not known to be normally distributed

  • 6.

    A carpenter wants to start building a new style of table, and surveys 100 people to identify an ideal size. He calculates a mean preferred length of 7.5 feet, with a standard deviation of 3 inches. If the carpenter was to make a confidence interval or conduct a hypothesis test to determine the ideal length of a table, should a z-distribution or a t-distribution be used?

    Answers:

    • A t-distribution should be used because the sample size is too small

    • A z-distribution should be used because the sample size is sufficiently large

    • A t-distribution should be used because the population standard deviation is not known

    • Neither table should be used because the data are not known to be normally distributed

  • 7.

    A jeweler carefully weighs and measures the diameter of 32 1-carat diamonds, the standard deviation for which is known to be 0.136mm. The jeweler tabulates a sample mean of 6.24mm, with a sample standard deviation of 0.124mm. If the jeweler was to make a confidence interval or conduct a hypothesis test, should a z-distribution or a t-distribution be used?

    Answers:

    • A t-distribution should be used because the population standard deviation is not known

    • A t-distribution should be used because the sample size is too small

    • Neither table should be used because the data are not known to be normally distributed

    • A z-distribution should be used because the sample size is sufficiently large

  • 8.

    A factory making down-stuffed pillows undergoes a quality control test, ensuring their pillows' weights are normally distributed with a standard deviation of .5 grams. The quality control tester samples 40 pillows to measure their weight. If the tester was to make a confidence interval or conduct a hypothesis test, should a z-distribution or a t-distribution be used?

    Answers:

    • A t-distribution should be used because the population standard deviation is not known

    • A t-distribution should be used because the sample size is too small

    • A z-distribution should be used because the sample size is sufficiently large

    • Neither table should be used because the data are not known to be normally distributed

  • 9.

    A lumberjack samples the diameters of 20 trees from a lot of 500 that are due to be harvested. The trees' mean diameter is calculated to be 25 inches with a sample standard deviation of 0.2 inches. If the lumberjack was to make a confidence interval or conduct a hypothesis test to determine whether all 500 trees are likely ready to be harvested, should a z-distribution or a t-distribution be used?

    Answers:

    • A t-distribution should be used because the population standard deviation is not known

    • Neither table should be used because the data are not known to be normally distributed

    • A z-distribution should be used because the sample size is sufficiently large

    • A t-distribution should be used because the sample size is too small

  • 10.

    A student, bored in class, decides to measure and tally each length of wood grain on her desk. She measures 193 lengths of grain, calculating a mean of 3.4 cm with a standard deviation of 1.2 cm. If the student was to make a confidence interval or conduct a hypothesis test, to determine the mean length of wood grain on all the desks in the class, should a z-distribution or a t-distribution be used?

    Answers:

    • A z-distribution should be used because the sample size is sufficiently large

    • Neither table should be used because the data are not known to be normally distributed

    • A t-distribution should be used because the population standard deviation is not known

    • A t-distribution should be used because the sample size is too small

  • 11.

    A pest controller is assessing outbreaks of cockroaches in a certain neighborhood in her district. Collecting data from 50 homes, she finds that 34% of them have a cockroach problem. She knows the population mean among neighborhoods in this district to be 15%. If she was to make a confidence interval or conduct a hypothesis test to determine if her sample is sound, should a z-distribution or t-distribution be used?

    Answers:

    • A t-distribution should be used because the sample size is too small

    • A z-distribution should be used because the sample size is sufficiently large

    • Neither table should be used because the data are not known to be normally distributed

    • A t-distribution should be used because the population standard deviation is not known

  • 12.

    A waiter is tabulating his tips for the past week and calculates a mean of $2.13 from a sample of 25 tables waited. He knows the standard deviation for his weekly tips since beginning work at this restaurant is $0.51. If he was to make a confidence interval or conduct a hypothesis test to determine how significant this sample was, should a z-distribution or t-distribution be used?

    Answers:

    • A z-distribution should be used because the sample size is sufficiently large

    • A t-distribution should be used because the population standard deviation is not known

    • Neither table should be used because the data are not known to be normally distributed

    • A t-distribution should be used because the sample size is too small

  • 13.

    A chiropractor is measuring a patient's spinal displacement across 33 vertebrae, calculating a mean displacement of 0.23cm. Among all her patients, she knows the standard deviation for mean spinal displacement is 0.04cm. If she was to make a confidence interval or conduct a hypothesis test, should a z-distribution or t-distribution be used?

    Answers:

    • A t-distribution should be used because the population standard deviation is not known

    • Neither table should be used because the data are not known to be normally distributed

    • A t-distribution should be used because the sample size is too small

    • A z-distribution should be used because the sample size is sufficiently large

  • 14.

    A standardized test writer collects a sample from 100 beta test takers, calculating their mean score as 28 out of 36, with a standard deviation of 5. The established standard deviation for scores on this test is known to be 2.5. With this information, if the test writer was to make a confidence interval or conduct a hypothesis test, should a z-distribution or t-distribution be used?

    Answers:

    • Neither table should be used because the data are not known to be normally distributed

    • A t-distribution should be used because the sample size is too small

    • A t-distribution should be used because the population standard deviation is not known

    • A z-distribution should be used because the sample size is sufficiently large

  • 15.

    A die maker rolls a 20-sided die (sides numbered 1 to 20) 100 times to test for balance, calculating a resulting mean of 11.1. By her standards, a sufficiently balanced 20-sided die rolls a mean of 10.5 with a standard deviation of 0.5. If she was to make a confidence interval or conduct a hypothesis test, should a z-distribution or t-distribution be used?

    Answers:

    • Neither table should be used because the data are not known to be normally distributed

    • A z-distribution should be used because the sample size is sufficiently large

    • A t-distribution should be used because the sample size is too small

    • A t-distribution should be used because the population standard deviation is not known

  • 16.

    A designer wants to use sequins on her next dress that are more reflective than the majority of all other sequins. She collects a random sample of 200 sequins and measures their reflectivity, finding a sample mean of 32%. Knowing the population standard deviation is 1.1%, if the designer was to make a confidence interval or conduct a hypothesis test, should a z-distribution or a t-distribution be used?

    Answers:

    • A z-distribution should be used because the sample size is sufficiently large

    • Neither table should be used because the data are not known to be normally distributed

    • A t-distribution should be used because the sample size is too small

    • A t-distribution should be used because the population standard deviation is not known

  • 17.

    When building a fence, you buy 50 planks of wood, and measure their widths, calculating a mean of 0.743 inches. The manufacturer of these planks states their width's standard deviation is 0.005 inches. If you were to make a confidence interval or conduct a hypothesis test, should a z-distribution or a t-distribution be used?

    Answers:

    • A t-distribution should be used because the population standard deviation is not known

    • Neither table should be used because the data are not known to be normally distributed

    • A z-distribution should be used because the sample size is sufficiently large

    • A t-distribution should be used because the sample size is too small

  • 18.

    A dog walker is trying to get their business started in a new city. They survey 20 dog owners inquiring for how long they each walk their dogs, coming up with a sample mean of 45 minutes. With data from their city, they determine the population standard deviation is 8.5 minutes. If they were to make a confidence interval or conduct a hypothesis test, should a z-distribution or a t-distribution be used?

    Answers:

    • A t-distribution should be used because the sample size is too small

    • A t-distribution should be used because the population standard deviation is not known

    • A z-distribution should be used because the sample size is sufficiently large

    • Neither table should be used because the data are not known to be normally distributed

  • 19.

    Out of curiosity, a cat groomer takes a sample of how much fur is left by each cat he grooms, by weight. The groomer weighs the brushed fur of 104 cats throughout the week, and calculates a mean of 12.7 grams. Sourcing from other groomers, he determines the population standard deviation to be 5.8 grams. If the groomer was to make a confidence interval or conduct a hypothesis test, should a z-distribution or a t-distribution be used?

    Answers:

    • A t-distribution should be used because the sample size is too small

    • A t-distribution should be used because the population standard deviation is not known

    • A z-distribution should be used because the sample size is sufficiently large

    • Neither table should be used because the data are not known to be normally distributed

  • 20.

    A wedding flower arranger tabulates 43 clients' purchases to determine her mean profit per client. She calculates a mean of $6043.34 with a standard deviation of $225.39. She hopes to compare this to other florists in her area. If she was to make a confidence interval or conduct a hypothesis test, should a z-distribution or a t-distribution be used?

    Answers:

    • Neither table should be used because the data are not known to be normally distributed

    • A t-distribution should be used because the population standard deviation is not known

    • A z-distribution should be used because the sample size is sufficiently large

    • A t-distribution should be used because the sample size is too small

  • 21.

    A carpenter wants to start building a new style of furniture, and surveys 100 people to identify an ideal style. He determines that the majority of people surveyed (63) prefer modern-style furniture. If the carpenter was to make a confidence interval or conduct a hypothesis test to determine the ideal length of a table, should a z-distribution or a t-distribution be used?

    Answers:

    • A t-distribution should be used because the sample size is too small

    • A t-distribution should be used because the population standard deviation is not known

    • A z-distribution should be used because the sample size is sufficiently large

    • Neither table should be used because the data are not known to be normally distributed

  • 22.

    A jeweler carefully weighs and measures the diameter of 45 1-carat diamonds to determine their worth. The jeweler tabulates a sample mean of 5.91mm, with a sample standard deviation of 0.093mm. If the jeweler was to make a confidence interval or conduct a hypothesis test, should a z-distribution or a t-distribution be used?

    Answers:

    • A t-distribution should be used because the population standard deviation is not known

    • Neither table should be used because the data are not known to be normally distributed

    • A t-distribution should be used because the sample size is too small

    • A z-distribution should be used because the sample size is sufficiently large

  • 23.

    A factory making down-stuffed pillows undergoes a quality control test, ensuring their pillows' weights are normally distributed with a standard deviation of .9 grams. The quality control tester samples 20 pillows to measure their weight. If the tester was to make a confidence interval or conduct a hypothesis test, should a z-distribution or a t-distribution be used?

    Answers:

    • A t-distribution should be used because the sample size is too small

    • Neither table should be used because the data are not known to be normally distributed

    • A t-distribution should be used because the population standard deviation is not known

    • A z-distribution should be used because the sample size is sufficiently large

  • 24.

    A lumberjack samples the diameters of 80 trees from a lot of 1000 that are due to be harvested. The trees' mean diameter is calculated to be 32 inches. The trees' diameter is known to have a standard deviation of 0.6 inches based on age. If the lumberjack was to make a confidence interval or conduct a hypothesis test to determine whether all 1000 trees are likely ready to be harvested, should a z-distribution or a t-distribution be used?

    Answers:

    • A t-distribution should be used because the sample size is too small

    • Neither table should be used because the data are not known to be normally distributed

    • A t-distribution should be used because the population standard deviation is not known

    • A z-distribution should be used because the sample size is sufficiently large

  • 25.

    A student, bored in class, decides to measure and tally each length of wood grain on her desk. She measures 49 lengths of grain, calculating a mean of 3.2. Since the desks are not real wood, all the veneer patterns are identical between desks. If the student was to make a confidence interval or conduct a hypothesis test, to determine the mean length of wood grain on all the desks in the class, should a z-distribution or a t-distribution be used?

    Answers:

    • Neither table should be used because the data are not known to be normally distributed

    • A z-distribution should be used because the sample size is sufficiently large

    • A t-distribution should be used because the sample size is too small

    • A t-distribution should be used because the population standard deviation is not known

  • 26.

    A pest controller is assessing outbreaks of cockroaches in a certain neighborhood in her district. Collecting data from 20 homes, she finds that they report to have a cockroach problem 44% of the time. She knows the standard deviation for percentage of time with cockroach problems among homes in this neighborhood to be 8%. If she was to make a confidence interval or conduct a hypothesis test, should a z-distribution or t-distribution be used?

    Answers:

    • A t-distribution should be used because the population standard deviation is not known

    • Neither table should be used because the data are not known to be normally distributed

    • A t-distribution should be used because the sample size is too small

    • A z-distribution should be used because the sample size is sufficiently large

  • 27.

    A waiter is tabulating his tips for the past week, and calculates a mean of $18.13 from a sample of 33 tables waited, with a standard deviation of $4.13. If he was to make a confidence interval or conduct a hypothesis test to determine how significant this sample was, should a z-distribution or t-distribution be used?

    Answers:

    • A t-distribution should be used because the sample size is too small

    • A z-distribution should be used because the sample size is sufficiently large

    • Neither table should be used because the data are not known to be normally distributed

    • A t-distribution should be used because the population standard deviation is not known

  • 28.

    A chiropractor is measures 37 of her patients' spinal displacements, calculating a mean displacement of 0.31cm. Among all her patients, she knows the standard deviation for mean spinal displacement is 0.07cm. If she was to make a confidence interval or conduct a hypothesis test, should a z-distribution or t-distribution be used?

    Answers:

    • A t-distribution should be used because the sample size is too small

    • A z-distribution should be used because the sample size is sufficiently large

    • Neither table should be used because the data are not known to be normally distributed

    • A t-distribution should be used because the population standard deviation is not known

  • 29.

    A standardized test writer collects a sample from 20 beta test takers, calculating their mean score as 31 out of 36, with a standard deviation of 2. The established standard deviation for scores on this test is known to be 2.2. With this information, if the test writer was to make a confidence interval or conduct a hypothesis test, should a z-distribution or t-distribution be used?

    Answers:

    • A t-distribution should be used because the sample size is too small

    • A t-distribution should be used because the population standard deviation is not known

    • A z-distribution should be used because the sample size is sufficiently large

    • Neither table should be used because the data are not known to be normally distributed

  • 30.

    A die maker rolls a 20-sided die (sides numbered 1 to 20) 20 times to test for balance, calculating a resulting mean of 10.5. By her standards, a sufficiently balanced 20-sided die rolls a mean of 10.5 with a standard deviation of 0.5. If she was to make a confidence interval or conduct a hypothesis test, should a z-distribution or t-distribution be used?

    Answers:

    • A z-distribution should be used because the sample size is sufficiently large

    • A t-distribution should be used because the population standard deviation is not known

    • A t-distribution should be used because the sample size is too small

    • Neither table should be used because the data are not known to be normally distributed

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