The image you uploaded contains two tasks: one regarding interval estimation for weights and the other about hypothesis testing for the mean height of students. I'll break down how to solve these tasks step by step.

Task 1: Interval Estimation for Weights

We are given a data set of weights:

$70, 65, 67, 72, 82, 56, 66, 76, 56, 45, 46, 54, 57, 76, 67, 87, 56, 65$

We need to estimate the population mean using the following confidence levels:

• 90%
• 95%

Steps for solving:

1. Calculate the sample mean ($\bar{x}$) and the sample standard deviation (s).
2. Find the Z-value corresponding to the confidence levels (for 90% and 95%).
3. Calculate the confidence intervals using the formula:

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

Where:

• $\bar{x}$ is the sample mean
• $s$ is the sample standard deviation
• $n$ is the sample size
• $Z$ is the Z-value from the Z-table corresponding to the confidence level.

Task 2: Hypothesis Testing for Heights

We are given a data set of heights:

$169, 164, 166, 171, 181, 155, 165, 175, 155, 144, 145, 153, 156, 175, 166, 186, 155, 164$

We need to test the hypothesis that the mean height is 170 cm, using the following confidence levels:

• 90%
• 95%

Steps for solving:

1. Set up the null hypothesis ($H_0$) and alternative hypothesis ($H_1$):

   • $H_0$: The mean height is 170 cm.
   • $H_1$: The mean height is not 170 cm.

2. Calculate the sample mean ($\bar{x}$) and sample standard deviation (s).

3. Perform a one-sample t-test since the sample size is small and we are assuming normality. Use the formula:

$t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$

Where:

• $\bar{x}$ is the sample mean
• $\mu_0$ is the hypothesized population mean (170 cm)
• $s$ is the sample standard deviation
• $n$ is the sample size

4. Find the critical t-value corresponding to the confidence levels (90% and 95%).
5. Compare the test statistic with the critical t-value to make a decision on whether to reject $H_0$.

Would you like me to compute these values for you or provide further details on how to approach these calculations?