To calculate the P-values for a t-test, we use the t-distribution and the degrees of freedom (df). The formula for degrees of freedom is generally $\text{df} = n - 1$, where $n$ is the sample size. Once we know the df and t-value, we can use a statistical calculator or software to determine the P-value for different types of tests:

(a) Two-tailed test, df = 9, t = 0.75

For a two-tailed test, the P-value represents the probability that the absolute value of the t-statistic is as extreme as or more extreme than the observed value (in either direction).

• Given:
  • $df = 9$
  • $t = 0.75$

Using a calculator:

• The P-value for a two-tailed test with these parameters is approximately 0.4731.

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(b) Upper-tailed test, df = 10, t = −0.70

For an upper-tailed test, we focus on the probability that the t-value is greater than or equal to the observed value. Since the given $t = -0.70$ is negative, the P-value in this case would be large because we’re interested in the area to the right of 0.

• Given:
  • $df = 10$
  • $t = -0.70$

Since it’s an upper-tailed test and the t-value is negative, we reflect it to the positive side (upper-tail always works with positive t-values).

• The P-value for this test is 0.7589.

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(c) Lower-tailed test, n = 20, t = −2.40

For a lower-tailed test, the P-value corresponds to the area to the left of the observed t-value (since we are focusing on smaller values for a lower-tailed test).

• Given:
  • $df = 20 - 1 = 19$
  • $t = -2.40$

Using a calculator:

• The P-value for a lower-tailed test with these parameters is approximately 0.0132.

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(d) Lower-tailed test, n = 20, t = −4.90

• Given:
  • $df = 20 - 1 = 19$
  • $t = -4.90$

Using a calculator:

• The P-value for a lower-tailed test with this extremely negative t-value is very small, approximately 0.0001.

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(e) Two-tailed test, n = 40, t = 1.50

For a two-tailed test with larger sample size:

• Given:
  • $df = 40 - 1 = 39$
  • $t = 1.50$

Using a calculator:

• The P-value for a two-tailed test is approximately 0.1413.

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Summary of P-values:

1. (a) Two-tailed test, $df = 9$, $t = 0.75$: P-value ≈ 0.4731
2. (b) Upper-tailed test, $df = 10$, $t = -0.70$: P-value ≈ 0.7589
3. (c) Lower-tailed test, $df = 19$, $t = -2.40$: P-value ≈ 0.0132
4. (d) Lower-tailed test, $df = 19$, $t = -4.90$: P-value ≈ 0.0001
5. (e) Two-tailed test, $df = 39$, $t = 1.50$: P-value ≈ 0.1413

Would you like more details on how to calculate these values or have any other questions?

Here are 5 relative questions to expand on:

1. What is the difference between one-tailed and two-tailed tests?
2. How does the degree of freedom affect the shape of the t-distribution?
3. How does the sample size (n) impact the t-statistic and P-value?
4. In what situations is it more appropriate to use a one-tailed vs a two-tailed test?
5. How can we interpret the P-value in hypothesis testing?

Tip: The smaller the P-value, the stronger the evidence against the null hypothesis.