Probability & Statistics for Machine Learning & Data Science (23)

▤ 목차

Confidence Intervals and Hypothesis testing

Quiz

728x90

Q1

Consider two sets of samples drawn from the same population that are randomly selected. Set X has a sample size = 10, and set Y has a sample size = 100. Which of the following statements is accurate about the confidence interval for the mean of the samples?

1. The confidence interval for set X is larger than the confidence interval for set Y.
2. The confidence interval for set X is smaller than the confidence interval for set Y.
3. The confidence interval for set X equals the confidence interval for set Y.
4. There isn't enough information to answer the question.

1

Confidence interval estimates the range within a population parameter, in this case, the population mean. Since set X has a smaller sample size, the sample distribution gets further from the normal distribution, and the standard deviation is larger—a larger standard deviation results in a larger confidence interval.

Q2

Suppose you have a sample of 100 heights of individuals from a specific population. Assume the population standard deviation is 1 cm, and the sample mean is 175cm from a random sample of 100 individuals. What expression describes the margin of error for a confidence level of 99%?

1. $z_{0.01}\cdot{1\over10}$
2. $z_{0.005}\cdot{1\over100}$
3. $z_{0.005}\cdot{1\over10}$
4. $z_{0.1}\cdot{1\over100}$

3

Since the question asks for a confidence level of 99% $\alpha/2={0.01\over2}=0.005$. The formula for the margin of error is $z_{\alpha/2}\cdot{\sigma\over\sqrt n}$

Q3

To calculate a confidence interval for the mean of a population, what assumptions must be made? Select all that apply.

 

1. The sample is random.
2. The population must follow a normal distribution.
3. The sample size must be big enough (usually over 30).
4. The sample must have a mean = 0 and a standard deviation = 1.

1

A random sample is crucial for a reliable confidence interval for the population mean as it ensures representativeness. Random sampling provides an equal chance for each population member to be included, reducing bias and enhancing

3

The Central Limit Theorem says the average sample mean converges to a normal distribution. It means that the bigger the sample, the closer it is to a normal, so we must ensure that there are enough points for this approximation to be good enough.

Q4

You have a sample size of 20 from a population with unknown mean and standard deviation. You measured that the sample mean is $\bar X=50$ and the sample standard deviation is $s=10$. A confidence level of 95% of confidence level is given by:

Hint: $t_{0.475}=2.093$

1. (48.95, 51.05)
2. (45.32, 54.68)
3. (45.2, 54.8)
4. (48.9, 51.1)

2

Q5

A manufacturing company takes a sample of 100 items in its product warehouse and determines that 22% of the sample contains a defect. Calculate the population margin of error with a 95% confidence interval.

Hint: $z_{\alpha/2}=1.96$

1. 0.0336
2. 0.0812
3. 0.0919
4. 0.3363

2

When calculating the confidence interval for proportions, the formula is

$$ \text{margin of error}=z_{a/2}\cdot\sqrt{{\hat p(1-\hat p)\over n}} $$

All the information provided is based on the Probability & Statistics for Machine Learning & Data Science | Coursera from DeepLearning.AI