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QUAN 2010 Notes Introduction to Business Statistics

Lauren M. Nelsen and Kimberly Swetz

Section 10.6 Test statistic Practice

Let’s practice deciding what test statistic to use in different situations!
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Exercise 10.6.1.

(a)

A retailer claims that \(70\%\) of customers are satisfied with their service. You want to test this claim using survey data from a random sample of 200 customers.
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  • \(z=\frac{\bar{p}-p}{\sqrt{\frac{p\cdot(1-p)}{n}}}\)
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  • \(t=\frac{\bar{x}-\mu}{(s/\sqrt{n})} \)
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  • \(z=\frac{\bar{x}-\mu}{(\sigma/\sqrt{n})} \)
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(b)

You want to test if the average time spent on a website is greater than 5 minutes. You look at a sample of visits to the website and find that in the sample, the average time spent was 5.5 minutes and you have data from that sample.
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  • \(z=\frac{\bar{p}-p}{\sqrt{\frac{p\cdot(1-p)}{n}}}\)
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  • \(t=\frac{\bar{x}-\mu}{(s/\sqrt{n})} \)
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  • \(z=\frac{\bar{x}-\mu}{(\sigma/\sqrt{n})} \)
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(c)

A credit card company claims that the average transaction amount is \(\$120\text{,}\) with a known population standard deviation of $30. A random sample of 50 recent transactions shows an average of $125. The company wants to know if there is evidence that the average transaction amount has increased.
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  • \(z=\frac{\bar{p}-p}{\sqrt{\frac{p\cdot(1-p)}{n}}}\)
    🔗
  • \(t=\frac{\bar{x}-\mu}{(s/\sqrt{n})} \)
    🔗
  • \(z=\frac{\bar{x}-\mu}{(\sigma/\sqrt{n})} \)
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(d)

A small business believes that the average delivery time to customers is 5 days. A recent audit of a random sample of 25 deliveries shows a sample mean of 5.4 days with a sample standard deviation of 1.2 days. The owner wants to know if the data suggests that delivery times have increased.
🔗
  • \(z=\frac{\bar{p}-p}{\sqrt{\frac{p\cdot(1-p)}{n}}}\)
    🔗
  • \(t=\frac{\bar{x}-\mu}{(s/\sqrt{n})} \)
    🔗
  • \(z=\frac{\bar{x}-\mu}{(\sigma/\sqrt{n})} \)
    🔗
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(e)

A digital marketing agency claims that at least \(40\%\) of users click on their ads. You sample 300 users, and 110 clicked. Test if the true proportion is less than \(40\%\text{.}\)
🔗
  • \(z=\frac{\bar{p}-p}{\sqrt{\frac{p\cdot(1-p)}{n}}}\)
    🔗
  • \(t=\frac{\bar{x}-\mu}{(s/\sqrt{n})} \)
    🔗
  • \(z=\frac{\bar{x}-\mu}{(\sigma/\sqrt{n})} \)
    🔗
🔗

(f)

A shipping company claims its packages arrive in 3 days on average, with a known population standard deviation of 0.5 days. You take a sample of 50 packages and want to test this claim.
🔗
  • \(z=\frac{\bar{p}-p}{\sqrt{\frac{p\cdot(1-p)}{n}}}\)
    🔗
  • \(t=\frac{\bar{x}-\mu}{(s/\sqrt{n})} \)
    🔗
  • \(z=\frac{\bar{x}-\mu}{(\sigma/\sqrt{n})} \)
    🔗
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Exercise 10.6.2.

(a)

A business professor suspects that students at her university spend less than 10 hours/week studying for statistics. She surveys 18 students, and the sample has a mean of 8.7 hours and a sample standard deviation of 2.5 hours.
🔗
  • \(z=\frac{\bar{p}-p}{\sqrt{\frac{p\cdot(1-p)}{n}}}\)
    🔗
  • \(t=\frac{\bar{x}-\mu}{(s/\sqrt{n})} \)
    🔗
  • \(z=\frac{\bar{x}-\mu}{(\sigma/\sqrt{n})} \)
    🔗
🔗

(b)

An accounting firm wants to know if the average number of hours worked per week by employees is different from the standard 40 hours. A random sample of 25 employees has a mean of 41.3 and a sample standard deviation of 2.8 hours.
🔗
  • \(z=\frac{\bar{p}-p}{\sqrt{\frac{p\cdot(1-p)}{n}}}\)
    🔗
  • \(t=\frac{\bar{x}-\mu}{(s/\sqrt{n})} \)
    🔗
  • \(z=\frac{\bar{x}-\mu}{(\sigma/\sqrt{n})} \)
    🔗
🔗

(c)

A clothing company believes that \(60\%\) of customers prefer shopping online. You collect a sample of 500 customers, and 275 say they prefer online shopping.
🔗
  • \(z=\frac{\bar{p}-p}{\sqrt{\frac{p\cdot(1-p)}{n}}}\)
    🔗
  • \(t=\frac{\bar{x}-\mu}{(s/\sqrt{n})} \)
    🔗
  • \(z=\frac{\bar{x}-\mu}{(\sigma/\sqrt{n})} \)
    🔗
🔗

(d)

A company advertises that its energy drink increases focus for an average of 120 minutes, and they know the population standard deviation is 10 minutes from large-scale lab testing. A sample of 40 users shows an average focus time of 117 minutes.
🔗
  • \(z=\frac{\bar{p}-p}{\sqrt{\frac{p\cdot(1-p)}{n}}}\)
    🔗
  • \(t=\frac{\bar{x}-\mu}{(s/\sqrt{n})} \)
    🔗
  • \(z=\frac{\bar{x}-\mu}{(\sigma/\sqrt{n})} \)
    🔗
🔗

(e)

A bank wants to test whether the average wait time at a new branch is different from the usual 5 minutes. A sample of 10 customers has a mean wait time of 6.2 minutes and a sample standard deviation of 1.1 minutes.
🔗
  • \(z=\frac{\bar{p}-p}{\sqrt{\frac{p\cdot(1-p)}{n}}}\)
    🔗
  • \(t=\frac{\bar{x}-\mu}{(s/\sqrt{n})} \)
    🔗
  • \(z=\frac{\bar{x}-\mu}{(\sigma/\sqrt{n})} \)
    🔗
🔗

(f)

A bakery wants to know if the average cost of an order has changed from $18. They sample 30 recent orders, and the sample standard deviation is \(\$2.50\text{.}\)
🔗
  • \(z=\frac{\bar{p}-p}{\sqrt{\frac{p\cdot(1-p)}{n}}}\)
    🔗
  • \(t=\frac{\bar{x}-\mu}{(s/\sqrt{n})} \)
    🔗
  • \(z=\frac{\bar{x}-\mu}{(\sigma/\sqrt{n})} \)
    🔗
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