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A researcher thinks that if knee surgery patients go to physical therapy three times a week (instead of 2 times), their recovery period will be shorter. Average recovery times for all knee surgery patients is 8.2 weeks (with PT twice a week). The researchers took a random sample of 34 patients who had knee surgery and asked them to do PT three times a week. The average recovery time of the 34 patients was 7.9 weeks with standard deviation 1.4 weeks.What is the test statistic?

Question

A researcher thinks that if knee surgery patients go to physical therapy three times a week (instead of 2 times), their recovery period will be shorter. Average recovery times for all knee surgery patients is 8.2 weeks (with PT twice a week). The researchers took a random sample of 34 patients who had knee surgery and asked them to do PT three times a week. The average recovery time of the 34 patients was 7.9 weeks with standard deviation 1.4 weeks.What is the test statistic?

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Solution

To calculate the test statistic, we will use the formula for a one-sample t-test, which is:

t = (X̄ - μ) / (s/√n)

where:

  • X̄ is the sample mean
  • μ is the population mean
  • s is the standard deviation of the sample
  • n is the sample size

In this case:

  • X̄ = 7.9 weeks (average recovery time of the 34 patients)
  • μ = 8.2 weeks (average recovery time for all knee surgery patients)
  • s = 1.4 weeks (standard deviation of the 34 patients)
  • n = 34 (number of patients in the sample)

Substituting these values into the formula, we get:

t = (7.9 - 8.2) / (1.4/√34) = -0.3 / (1.4/√34)

Calculating the denominator first:

1.4/√34 = 0.24 (approximately)

Then substitute this into the equation:

t = -0.3 / 0.24 = -1.25

So, the test statistic (t) is approximately -1.25.

This problem has been solved

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