To calculate the test statistic for this data set, we need to follow these steps:

1. Organize the data

We have the following data for each category (clothes, food, and toys):

• Clothes: 28, 16, 51, 35, 28, 31, 17, 31, 20 (9 data points)
• Food: 38, 41, 42, 52, 47, 42, 34, 43, 57, 47, 44, 54 (12 data points)
• Toys: 53, 44, 48, 56, 63, 53, 48, 58, 47, 51, 51 (11 data points)

2. Calculate the sample means for each group

Let’s calculate the sample mean of each category.

3. Hypothesized mean

We need to subtract the hypothesized mean from the sample means. You haven’t provided the hypothesized mean, so I’ll assume it’s a value you have or should have from the context of the problem.

4. Calculate the standard error of the mean

The standard error of the mean is calculated as:

$SE = \frac{s}{\sqrt{n}}$

where $s$ is the standard deviation of the sample and $n$ is the number of data points.

5. Test statistic

The test statistic is given by:

$t = \frac{\text{Sample mean} - \text{Hypothesized mean}}{SE}$

Let me calculate this step by step using Python for accuracy.Here are the test statistics for each category:

1. Clothes:

   • Sample Mean: 28.5556
   • Standard Deviation: 10.7367
   • Standard Error: 3.5789
   • Test Statistic: $-3.1977$

2. Food:

   • Sample Mean: 45.0833
   • Standard Deviation: 6.6805
   • Standard Error: 1.9285
   • Test Statistic: $2.6359$

3. Toys:

   • Sample Mean: 52.0000
   • Standard Deviation: 5.4589
   • Standard Error: 1.6459
   • Test Statistic: $7.2907$

These test statistics compare each sample mean to the hypothesized mean of 40 (which was assumed).

Would you like more details or clarification on any part? Here are five related questions to further explore:

1. How do you interpret the negative test statistic for the clothes data?
2. What does the large test statistic for the toys data suggest?
3. How does changing the hypothesized mean affect the test statistic?
4. Can you apply a different confidence level to these results?
5. What are the implications of the standard error for each group?

Tip: Always check the degrees of freedom when using a t-test for small sample sizes, as it affects the critical values for the test statistic.