To compute the test statistic for the paired $t$-test, we follow these steps:

Step 1: Define the differences ($d$) and compute the sample statistics.

For each golfer, calculate the difference: $d = \text{Score (new design)} - \text{Score (old design)}.$

Scores:

Golfer	Old Score ($X_{\text{old}}$)	New Score ($X_{\text{new}}$)	Difference ($d$)
1	83	77	$77 - 83 = -6$
2	86	90	$90 - 86 = 4$
3	86	83	$83 - 86 = -3$
4	93	87	$87 - 93 = -6$
5	80	81	$81 - 80 = 1$
6	89	84	$84 - 89 = -5$
7	94	91	$91 - 94 = -3$
8	80	77	$77 - 80 = -3$

Differences ($d$):

$d = [-6, 4, -3, -6, 1, -5, -3, -3]$

Compute the mean and standard deviation of $d$:

$\bar{d} = \frac{\sum d}{n} = \frac{-6 + 4 - 3 - 6 + 1 - 5 - 3 - 3}{8} = \frac{-21}{8} = -2.625$

$s_d = \sqrt{\frac{\sum (d_i - \bar{d})^2}{n-1}}$ We calculate the squared differences:

• For $d_1 = -6, (-6 - -2.625)^2 = (-3.375)^2 = 11.390625$
• For $d_2 = 4, (4 - -2.625)^2 = (6.625)^2 = 43.890625$
• For $d_3 = -3, (-3 - -2.625)^2 = (-0.375)^2 = 0.140625$
• For $d_4 = -6, (-6 - -2.625)^2 = (-3.375)^2 = 11.390625$
• For $d_5 = 1, (1 - -2.625)^2 = (3.625)^2 = 13.140625$
• For $d_6 = -5, (-5 - -2.625)^2 = (-2.375)^2 = 5.640625$
• For $d_7 = -3, (-3 - -2.625)^2 = (-0.375)^2 = 0.140625$
• For $d_8 = -3, (-3 - -2.625)^2 = (-0.375)^2 = 0.140625$

Sum of squared differences: $\sum (d_i - \bar{d})^2 = 11.390625 + 43.890625 + 0.140625 + 11.390625 + 13.140625 + 5.640625 + 0.140625 + 0.140625 = 85.875$

Standard deviation: $s_d = \sqrt{\frac{85.875}{8-1}} = \sqrt{\frac{85.875}{7}} = \sqrt{12.268} = 3.503$

Step 2: Compute the test statistic $t$:

$t = \frac{\bar{d} - 0}{s_d / \sqrt{n}} = \frac{-2.625 - 0}{3.503 / \sqrt{8}} = \frac{-2.625}{3.503 / 2.828} = \frac{-2.625}{1.239} = -2.119$

Final Answer:

The test statistic is: $t = -2.119$

Would you like to proceed with the p-value or interpretation?

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Questions for exploration:

1. How is the significance level ($\alpha = 0.05$) used in this hypothesis test?
2. Why do we use a paired $t$-test instead of an independent $t$-test here?
3. What assumptions are required for the paired $t$-test?
4. How does the sample size ($n = 8$) affect the reliability of the test?
5. How would the results change if the standard deviation of $d$ were smaller?

Tip: Always check whether the data meets the assumptions of normality when using parametric tests like the ( t