I cannot directly extract text or data from images without processing them. Let me extract the necessary information from the image and solve the problem using Python.from sympy import * import math

Given values from the problem

n = 78 # sample size mean = 102.38 # sample mean std_dev = 9.7 # sample standard deviation confidence_level = 0.98 # 98% confidence

Calculate the critical z-value for a 98% confidence interval (two-tailed)

alpha = 1 - confidence_level z_critical = abs(stats.norm.ppf(alpha / 2))

Calculate the margin of error

margin_of_error = z_critical * (std_dev / math.sqrt(n))

Calculate confidence interval

lower_bound = mean - margin_of_error upper_bound = mean + margin_of_error

lower_bound, upper_bound--------------------------------------------------------------------------- NameError Traceback (most recent call last) Cell In[1], line 12 10 # Calculate the critical z-value for a 98% confidence interval (two-tailed) 11 alpha = 1 - confidence_level ---> 12 z_critical = abs(stats.norm.ppf(alpha / 2)) 14 # Calculate the margin of error 15 margin_of_error = z_critical * (std_dev / math.sqrt(n))

NameError: name 'stats' is not defined failed_with_in_kernel_exceptionIt seems I made a minor error with an undefined library. Let me correct this and compute the solution.import scipy.stats as stats import math

Given values from the problem

n = 78 # sample size mean = 102.38 # sample mean std_dev = 9.7 # sample standard deviation confidence_level = 0.98 # 98% confidence

Calculate the critical z-value for a 98% confidence interval (two-tailed)

alpha = 1 - confidence_level z_critical = abs(stats.norm.ppf(alpha / 2))

Calculate the margin of error

margin_of_error = z_critical * (std_dev / math.sqrt(n))

Calculate confidence interval

lower_bound = mean - margin_of_error upper_bound = mean + margin_of_error

lower_bound, upper_bound(99.82495183692868, 104.93504816307131)(99.82495183692868, 104.93504816307131)successThe 98% confidence interval for the mean ($\mu$) is:

• Lower Bound: 99.8250
• Upper Bound: 104.9350

This indicates that the population mean likely falls within this interval.

For the hypothesis testing:

1. If $\mu = 100$ (null hypothesis), the population mean lies within the confidence interval.
2. If $\mu \neq 100$ (alternative hypothesis), the population mean is outside the interval.

Thus, the null hypothesis $H_0: \mu = 100$ appears to be correct.

Would you like a deeper explanation, or do you have additional questions? Here are five related questions:

1. How is the critical value $z$ calculated for different confidence levels?
2. What is the significance of the confidence interval in hypothesis testing?
3. How does sample size affect the confidence interval width?
4. What happens to the interval if the standard deviation increases?
5. Can this approach be used for small samples (e.g., $n < 30$)? Why or why not?

Tip: Always check whether the population standard deviation or the sample standard deviation is provided to apply the correct formula.