Module 4: Confidence Intervals

Q: What is a key difference between a point estimate and an interval estimate?

• A point estimate uses a single value to estimate a population parameter; an interval estimate uses a range of values to estimate a population parameter.
• A point estimate uses a range of values to estimate a sample statistic; an interval estimate uses a single value to estimate a sample statistic.
• A point estimate uses a range of values to estimate a population parameter; an interval estimate uses a single value to estimate a population parameter.
• A point estimate uses a single value to estimate a sample statistic; an interval estimate uses a range of values to estimate a sample statistic.

Explanation: This distinction is a reflection of the fact that point estimates provide a specific numerical value as an estimate of a parameter (for example, a sample mean can estimate the mean of a population), whereas interval estimates provide a range of values within which the true parameter value is likely to fall, along with a level of confidence associated with that interval.

Q: A data professional working for a moving company is estimating the average time it takes to complete a move. Based on a sample mean of 3 hours, they construct the following 95% confidence interval: [2.5 , 3.5]. What does 95% refer to?

• The percentage of all possible sample means that fall within the range of the interval
• The success rate of the estimation process 
• The margin of error
• The percentage of data values in the dataset

Explanation: Consequently, the right interpretation of the 95% in this context is the success rate of the estimate procedure. This means that the real population parameter would be caught in 95% of the intervals formed from samples during the estimating process.

Q: A data analytics team with a clothing manufacturer constructs a confidence interval to help estimate future returns. First, they identify the sample statistic. Then, they choose a confidence level of 95%. According to the four steps to constructing a confidence interval for a proportion, what should they do next?

• Plot a histogram
• Choose a confidence level
• Calculate the interval
• Find the margin of error 

Explanation: This includes calculating the standard error and then multiplying it by the relevant critical value from the standard normal distribution (or using the t-distribution for small samples) to evaluate the extent to which the sample percentage may differ from the actual proportion of the population.

Q: A data professional working for a light bulb manufacturer is estimating the mean bulb lifespan based on sample data. They construct a 95% confidence interval using a sample size of 100. In addition, they construct a 95% confidence interval using a sample size of 1,000. What happens as the sample size increases?

• The margin of error decreases.
• The margin of error increases.
• The population parameter gets larger.
• The confidence interval gets wider.

Explanation: The size of the sample and the degree of confidence that is selected both have an impact on the margin of error that is included in a confidence interval. Increasing the size of the sample population results in a reduction in the standard error of the mean, which is a component of the margin of error calculation. As the standard error decreases, the estimation of the population parameter (such as the average bulb lifetime) gets more accurate. This is because the standard error is less. As a consequence of this, the margin of error, which is a measure of the accuracy of the estimate, lowers as the sample size grows.

Q: What argument of the scipy.stats.norm.interval() function can be used to choose the confidence level?

• Alpha 
• scale
• std
• loc 

Explanation: It is possible to construct a confidence interval by using the interval() function when dealing with statistical distributions such as the normal distribution. Within the context of this function, the alpha parameter is a representation of the significance level or the complement of the confidence level, which is equal to one minus the confidence level. Therefore, if you wish to generate a confidence interval with a 95% probability, you would normally set alpha to 0.05. This is since the confidence level is equal to 1 minus alpha, which equals 0.95.

Q: Fill in the blank: Because there is more uncertainty involved in estimating the standard error, data professionals use the _____ when working with a small sample size.

• s-distribution
• normal distribution
• t-distribution 
• z-distribution

Explanation: It is common practice to use the sample standard deviation to estimate the population standard deviation when the sample size is small. This practice introduces extra uncertainty. More accurately than the normal (z) distribution, the t-distribution takes into account this uncertainty. This is particularly true in situations when the sample sizes are small and the standard deviation of the population is unknown. The t-distribution becomes closer and closer to mimicking the normal distribution as the sample size grows.

Q: At what sample size does the t-distribution become practically the same as the normal distribution? 

• 10
• 5
• 1
• 30 

Explanation: As the volume of the sample rises, the t-distribution approaches the z-distribution, which is the ordinary normal distribution, in a very close manner. In general, when the sample size is approximately thirty or more, the differences between the t-distribution and the normal distribution become insignificant for the purposes of practical application. The reason for this is that as the size of the sample rises, the standard deviation of the sample becomes a more accurate approximation of the standard deviation of the population. This results in a reduction in the amount of uncertainty that the t-distribution takes into consideration.

Q: What would a data professional use to estimate a population parameter using a range of values?

• Interval estimate 
• Point estimate
• Z-score
• Sampling frame

Explanation: An interval estimate gives a range of values within which the real population parameter is predicted to reside, as well as a degree of confidence associated with that interval. In addition, an interval estimate includes confidence levels. In contrast to a point estimate, which provides an estimate of a population parameter based on a single value, this method employs many values. It is standard practice in the field of statistics to make use of interval estimates to give a more nuanced view of the uncertainty associated with parameter estimations. This means that interval estimates take into consideration the variability in data and sampling.

Q: What concept describes the likelihood that a particular sampling method will produce a confidence interval that includes the population parameter?

• Confidence level 
• Margin of error
• Sample statistic
• Point estimate

Explanation: A sampling method's confidence interval is the likelihood that it will include the real population parameter. The confidence level is the probability that this will occur. A usual way to describe it is as a percentage (for example, a confidence level of 95%).In repeated sampling, a higher degree of confidence indicates that there is a larger possibility that the confidence interval accurately represents the underlying population parameter.

Q: A data professional working for a media company is estimating the average amount of time a visitor spends on their website. Based on a sample mean of 4 minutes, they construct the following 95% confidence interval: [3.8 , 4.2]. What does 95% refer to?

• The margin of error
• The percentage of all possible sample means that fall within the range of the interval
• The percentage of data values in the dataset
• The success rate of the estimation process 

Explanation: A confidence level of 95% indicates that if we were to repeatedly sample from the population and construct confidence intervals in the same manner, approximately 95% of those intervals would contain the true population parameter (in this case, the true average amount of time a visitor spends on the website). This is the case because the confidence intervals would cover the entire population. The value of this variable represents the degree of certainty or likelihood that the interval [3.8, 4.2] accurately represents the mean of the population. This interpretation places more of an emphasis on the dependability of the estimating process as opposed to the margin of error, the percentage of sample means, or the percentage of data values included inside the dataset.

Q: According to the four steps that detail how to construct a confidence interval for a proportion, which of the following activities are involved in this process? Select all that apply.

• Plot a histogram
• Choose a confidence level 
• Find the margin of error 
• Calculate the interval 

Explanation: This is what determines the chance that the confidence interval will include the actual percentage of the population. To do this, you will need to compute the standard error and then use it to ascertain the range of values around the sample percentage that most likely includes the same proportion as the actual population.To compute the confidence interval itself, this step makes use of the sample percentage as well as the margin of error.

Q: A data professional is using scipy.stats.norm.interval() in Python to construct a confidence interval. Which of the following pieces of code can they use to choose a confidence level of 99%?

1. scale = 0.99
2. std = 0.99
3. alpha = 0.99 
4. loc = 0.99

Explanation: The alpha argument in the scipy.stats.norm.interval(alpha, loc, scale) function is responsible for determining the degree of confidence. Adjusting the alpha parameter to the required degree of confidence in the form of a decimal fraction is necessary. The value of alpha would be 0.99 if the degree of confidence was 99%. It is possible to choose not to use the other parameters, which include the scale for the standard deviation and the location for the mean, depending on the nature of the issue.

Q: A data professional working for a theme park is estimating the mean time visitors spend in the park. They construct the following 95% confidence interval based on a sample mean of 3.5 hours: [2.5, 4.5]. What is the margin of error?

• +/- 4.5 hours
• +/- 1 hour 
• +/- 2.5 hours
• +/- 2 hours

Q: Which of the following statements accurately describe the graph of the t-distribution? Select all that apply.

• It has smaller tails than the standard normal distribution.
• As the sample size decreases, the t-distribution approaches the normal distribution.
• It has larger tails than the standard normal distribution. 
• As the sample size increases, the t-distribution approaches the normal distribution. 

Explanation: This assertion is accurate because the t-distribution gets increasingly comparable to the conventional normal distribution (z-distribution) as the sample size grows. The statement in question is correct. For example, in comparison to the usual normal distribution (z-distribution), the t-distribution has heavier tails. This indicates that the t-distribution has a bigger probability in the tails, which reflects a greater degree of uncertainty owing to the lower sample sizes.

Q: Which of the following statements accurately describe a point estimate? Select all that apply.

• A point estimate estimates a sample statistic.
• A point estimate uses a range of values.
• A point estimate estimates a population parameter. 
• A point estimate uses a single value. 

Q: In the context of constructing a confidence interval of a population mean, what does the loc argument of the scipy.stats.norm.interval() function refer to?

• Sample standard error
• Sample mean 
• Interquartile range
• Confidence level

Explanation: Utilizing the normal distribution as a foundation, the scipy.stats.norm.interval() method is responsible for constructing a confidence interval. Three inputs are required: alpha, which represents the amount of confidence, loc, which represents the mean, and scale, which represents the standard deviation. In the context of creating a confidence interval for a population mean, the term "loc" refers to the sample mean in particular. This is because the interval is centered on this point estimate.

Q: What shape is the graph of the t-distribution?

• Rectangular shape
• Circular shape
• Square shape
• Bell shape 

Explanation: It is the degrees of freedom parameter that determines the form of the bell curve of the t-distribution. When the degrees of freedom are lower, the bell curve's shape differs from that of the normal distribution in that it is more spread out (has heavier tails).

Q: A data analytics team at a book publisher researches the most popular book subject matter based on sample data. They construct a 95% confidence interval using a sample size of 250. They also construct a 95% confidence interval using a sample size of 5,000. What happens as the sample size increases?

• The confidence interval gets wider.
• The population parameter gets larger.
• The margin of error decreases. 
• The margin of error increases.

Explanation: One of the factors that might affect the margin of error in a confidence interval is the size of the sample. More specifically, the magnitude of the estimate's standard error reduces in proportion to the size of the sample. In other words, a decreasing standard error indicates that the estimation of the population parameter gets more accurate.Because of this, the confidence interval will get increasingly narrow as the sample size increases. This is because we will be able to estimate the population parameter with a greater level of precision. As a consequence of this, the margin of error, which is a measure of the accuracy of the estimate, lowers as the sample size grows.

Q: A data professional at an electricity utility works on a project involving household demand based on sample data. They want to construct a 95% confidence interval using a sample size of 5,000. However, they are unable to get enough data. So they decide to construct a 95% confidence interval using a sample size of 500. What happens as a result of this smaller sample size?

• The margin of error will decrease.
• The population parameter will get larger.
• The confidence interval will get narrower.
• The margin of error will increase. 

Explanation: An inverse relationship exists between the square root of the sample size and the margin of error that is associated with a confidence interval. Consequently, the margin of error will reduce as the sample size grows, and vice versa. This is the case. When compared to 5,000, the sample size of 500 is much less, which means that the estimation of the population parameter (such as household demand) will be subject to less precision. This leads to a wider margin of error, which indicates that there is a greater degree of uncertainty in the estimation of the population parameter.