Confidence Intervals and, Point Estimations are important concepts in statistics, which are applied widely in industries such as data analysis, research, and many others. Confidence intervals give a range of values of the population characteristic while point estimates give a single value that best represents the population parameter.

In this article, we give a comprehensive understanding of how to find point estimates from confidence intervals, the significance of point estimates, and examples to make the understanding easy and offer clarity.

What is Confidence Interval?

A confidence interval is an interval of values calculated from the sample observations expected to contain the population parameter. It is usually stated to some level of confidence, for instance, 95% or 99% which defines the level of confidence that indicates the degree of certainty associated with the interval.

Confidence Interval Formula

[Tex]CI = \bar{x} \pm Z \left( \frac{\sigma}{\sqrt{n}} \right)[/Tex]

where:

• [Tex]\bar{x}[/Tex] is Sample Mean
• [Tex]Z[/Tex] is the Z-score corresponding to Desired Confidence Level
• [Tex]\sigma[/Tex] is Population Standard Deviation
• [Tex]n[/Tex] is Sample Size

Example: Z-score for a 95% confidence level is about 1. 96. If the sample mean is 50, the population standard deviation is 10, and the sample size is 100, the confidence interval can be calculated as:

[Tex]CI = 50 \pm 1.96 \left( \frac{10}{\sqrt{100}} \right) = 50 \pm 1.96[/Tex]

Thus the confidence interval will be equal to (48.04, 51.96).

What is a Point Estimate?

A Point Estimate can be described as an single value which is used to estimate a parameter of the population.

Two most frequently occurring point estimates include the sample mean – used in estimating the population mean as well as the sample proportion useful for estimating the population proportion.

Example: Upon administering a question to 100 respondents and getting 60 of them to choose a specific brand, 0.60 is the sample proportion which is the point estimate of the population proportion.

Relationship Between Confidence Intervals and Point Estimates

Confidence intervals and point estimates are indeed, cousins or to use a statistical language, they are complementary to each other. The point estimate is the middle value in the confidence interval and gives the best single value estimate of the population parameter.

While, the confidence interval offers a range that is likely to contain the true parameter, thus offering some sort of reliability and precision of the statistic.

How to Calculate a Point Estimate from a Confidence Interval

If you need to derive a point estimate from a confidence interval, it can be calculated using the middle point of the interval. Given a confidence interval (L, U):

Formula:

[Tex]\text{Point Estimate} = \frac{L + U}{2}[/Tex]

where:

• L is Lower Bound
• U is Upper Bound of Confidence Interval

Example: If the confidence interval is (48.04, 51.96), the point estimate can be calculated as:

[Tex]\text{Point Estimate} = \frac{48.04 + 51.96}{2} = 50[/Tex]

Examples on Point Estimate from Confidence Interval

Example 1: Confidence Interval for a Mean, say there is a set of 50 students; they took an achievement test and got a mean test score of 75 with a standard deviation of 10. Calculate the 95% level of confidence interval for the mean of the test scores.

Solution:

[Tex]CI = 75 \pm 1.96 \left( \frac{10}{\sqrt{50}} \right) = 75 \pm 2.77[/Tex]

Confidence interval is (72.23, 77.77). The point estimate is:

[Tex]\text{Point Estimate} = \frac{72.23 + 77.77}{2} = 75[/Tex]

Example 2: Confidence Interval for a Proportion: A survey of 200 people finds that 150 support a new policy. Calculate the 95% confidence interval for the proportion of people supporting the policy.

Solution:

[Tex]\hat{p} = \frac{150}{200} = 0.75[/Tex]

[Tex]CI = 0.75 \pm 1.96 \sqrt{\frac{0.75(1 – 0.75)}{200}} = 0.75 \pm 0.06[/Tex]

Confidence interval is (0.69, 0.81). The point estimate is:

[Tex]\text{Point Estimate} = \frac{0.69 + 0.81}{2} = 0.75[/Tex]

Practice Questions on Point Estimate from Confidence Interval

Question 1. A 95% confidence interval for the mean height of students in a school is (150 cm, 170 cm). What is the point estimate of the mean height?

Question 2. A 90% confidence interval for the average weight of apples in a basket is (100 grams, 120 grams). Find the point estimate of the average weight.

Question 3. A 95% confidence interval for the mean time it takes to complete a task is (30 minutes, 40 minutes). What is the point estimate of the mean time?

Question 4. The 90% confidence interval for the mean monthly expenditure on groceries is ($250, $350). Find the point estimate for the mean monthly expenditure.

Common Mistakes on Point Estimate from Confidence Interval

• Misinterpreting the Confidence Level: The confidence level, for instance, 95%, is not the probability that the interval includes the population parameter. Instead of this, it means that if one were to sample many times and construct intervals in the same way, then 95% of such intervals, would indeed cover the true parameter.

• Ignoring Sample Size: More sample size results to decreased interval width, and thus gives increased accuracy of estimates.

• Assuming Normal Distribution: The formulas appears to be based on the normality which may not be appropriate for small samples or non-normal data.

Conclusion

In conclusion, finding the point estimate from a confidence interval is one of the simplest yet one of the most essential activities in statistics. Confidence interval and point estimates are important concepts that when understood you can always be in a position to comprehend data and make the right decisions.

In addition to the theoretical presentation of the concepts listed above, the examples and problems incorporated enable the user to appreciate the significance of these concepts in statistical analysis. The students and professionals by learning these concepts can improve the quality of their logical thinking and, as a result, contribute to higher-quality research.

Read More:

• Point Estimation
• Confidence Interval

FAQs on Point Estimate from Confidence Interval

What is the main purpose of a point estimate?

The reason for point estimation is to have an best single value estimate of a population parameter based on sample data.

How does sample size affect confidence intervals?

Confidence intervals becomes more narrow when larger sample sizes are used in the computation process.

Can confidence intervals be asymmetric?

Yes, it is possible for the confidence intervals to be asymmetric, primarily for the proportions or non-normal data.

Why is the midpoint of the confidence interval used as the point estimate?

It is the central value of the interval and gives the best single point estimate of the population parameter.

How is the margin of error related to the confidence interval?

The margin of error is half the width of the confidence interval ; this means that it is the the maximum expected difference between the point estimate and the true parameter.

Can confidence intervals be used for non-quantitative data?

Confidence intervals are mainly used with quantitative data although similar concept can be applied to categorical data through proportions.