To calculate the 95% confidence interval for the average time spent studying per week, we need to first calculate the t-score using the given information. The formula to calculate the t-score for a confidence interval is:

$t = \frac{X - \mu}{\frac{s}{\sqrt{n}}}$

Where:

• $X$ = sample mean = 12 hours
• $s$ = sample standard deviation = 3 hours
• $n$ = sample size = 18 students
• $\mu$ = population mean (which is unknown, and we're estimating it)

For a confidence interval, the t-score is found from the t-distribution table based on the degrees of freedom (df). The degrees of freedom are calculated as:

$df = n - 1 = 18 - 1 = 17$

The t-value for a 95% confidence interval with 17 degrees of freedom can be looked up in the t-distribution table (or calculated using software). For a 95% confidence level, the t-value is approximately 2.109 (this is a one-tailed value since we are interested in a 95% interval, which leaves 5% total in both tails).

So, the t-score is 2.109.

Now, let's calculate the 95% confidence interval for the average time spent studying per week.

The formula for the confidence interval is:

$\text{CI} = \bar{x} \pm t \times \frac{s}{\sqrt{n}}$

Substituting the values:

• Sample mean $\bar{x} = 12$
• Standard deviation $s = 3$
• Sample size $n = 18$
• t-score $t = 2.109$

$\text{Margin of Error} = 2.109 \times \frac{3}{\sqrt{18}} = 2.109 \times 0.707 = 1.49$

Thus, the 95% confidence interval is:

$12 \pm 1.49 \quad \text{or} \quad (10.51, 13.49)$

So, the 95% confidence interval for the average time spent studying per week is (10.51 hours, 13.49 hours).

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Do you need any further details on how to interpret this confidence interval, or do you have any other questions?

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Here are 5 related questions:

1. How would you calculate a confidence interval if the population standard deviation was known?
2. What would happen to the confidence interval if the sample size was increased?
3. How do we interpret a 95% confidence interval in practical terms?
4. Why do we use the t-distribution instead of the z-distribution here?
5. How would the confidence interval change if the sample standard deviation was larger?

Tip: Increasing the sample size generally reduces the margin of error, making the confidence interval narrower and the estimate more precise.