Critical Value Of 90 Confidence Interval

The critical value is a cornerstone in hypothesis testing and confidence interval construction, acting as a threshold against which test statistics are compared to determine the statistical significance of observed results. When constructing a 90% confidence interval, understanding the critical value is paramount for accurately defining the range within which a population parameter is likely to fall. This article delves into the concept of critical values, particularly focusing on their application in creating 90% confidence intervals.

Understanding Critical Values

A critical value is a point on the scale of the test statistic beyond which we reject the null hypothesis. It reflects the level of significance (alpha, denoted as α) chosen for the test. In simpler terms, it tells us how extreme our sample statistic needs to be before we can confidently say that the null hypothesis is unlikely to be true.

• Significance Level (α): The probability of rejecting the null hypothesis when it is actually true (Type I error).
• Confidence Level: The probability that the interval estimate contains the population parameter. For a 90% confidence interval, the significance level (α) is 10% (1 - 0.90 = 0.10).

The critical value is derived from the sampling distribution of the test statistic and depends on the chosen significance level (α) and whether the test is one-tailed or two-tailed.

Critical Values and Confidence Intervals

Confidence intervals provide a range of values within which we estimate a population parameter lies with a certain level of confidence. The general form of a confidence interval is:

Confidence Interval = Sample Statistic ± (Critical Value * Standard Error)

Here's how each component plays its part:

• Sample Statistic: This is the point estimate of the population parameter calculated from the sample data (e.g., sample mean).
• Critical Value: Determines the width of the confidence interval. A larger critical value results in a wider interval, indicating a greater level of uncertainty.
• Standard Error: Measures the variability of the sample statistic.

Determining the Critical Value for a 90% Confidence Interval

The process of finding the critical value differs slightly depending on whether you're dealing with a z-distribution (for large samples or known population standard deviation) or a t-distribution (for small samples or unknown population standard deviation).

1. Using the Z-Distribution

The z-distribution is a standard normal distribution with a mean of 0 and a standard deviation of 1. It's used when the sample size is large enough (typically n > 30) or when the population standard deviation is known.

• For a 90% confidence interval, α = 0.10. Since the confidence interval is two-tailed (we're interested in both ends of the distribution), we divide alpha by 2: α/2 = 0.05.
• We then look for the z-score that corresponds to the cumulative probability of 1 - α/2 = 0.95. In other words, we want the z-score that leaves 5% in the upper tail of the distribution.
• Using a z-table or a statistical calculator, the z-score corresponding to 0.95 is approximately 1.645.

Therefore, the critical value for a 90% confidence interval using the z-distribution is 1.645.

2. Using the T-Distribution

The t-distribution is similar to the z-distribution but has heavier tails, reflecting the added uncertainty when the population standard deviation is unknown and estimated from the sample. The t-distribution is characterized by its degrees of freedom (df), which are typically calculated as n - 1, where n is the sample size.

• For a 90% confidence interval, α = 0.10. Again, since the confidence interval is two-tailed, we divide alpha by 2: α/2 = 0.05.
• We need to find the t-value that corresponds to a cumulative probability of 0.95 with the appropriate degrees of freedom (df = n - 1).
• Using a t-table or a statistical calculator, locate the t-value for α/2 = 0.05 and the corresponding degrees of freedom. The critical t-value will depend on the sample size.

Example: If the sample size is 25, then df = 25 - 1 = 24. Looking up the t-value for α/2 = 0.05 and df = 24 in a t-table, we might find a critical value of approximately 1.711.

Key Differences and Considerations:

• Sample Size: For large sample sizes (generally n > 30), the t-distribution approaches the z-distribution. In such cases, the critical values from both distributions will be very similar. However, for smaller sample sizes, the t-distribution provides a more accurate critical value due to its heavier tails.
• Degrees of Freedom: Always remember to use the correct degrees of freedom (n - 1) when using the t-distribution.
• Choosing Between Z and T: If the population standard deviation is known, use the z-distribution. If the population standard deviation is unknown and estimated from the sample, use the t-distribution.

Calculating the 90% Confidence Interval: Examples

Let's illustrate how to calculate a 90% confidence interval using both the z-distribution and the t-distribution.

Example 1: Using the Z-Distribution

Suppose we want to estimate the average height of adult women in a city. We collect a random sample of 100 women and find that the sample mean height is 64 inches, and the population standard deviation is known to be 2.5 inches.

1. Sample Statistic: Sample mean (x̄) = 64 inches
2. Population Standard Deviation (σ): 2.5 inches
3. Sample Size (n): 100
4. Critical Value (zα/2): 1.645 (for a 90% confidence interval)
5. Standard Error: σ / √n = 2.5 / √100 = 0.25

Now, we can calculate the 90% confidence interval:

Confidence Interval = x̄ ± (zα/2 * Standard Error) Confidence Interval = 64 ± (1.645 * 0.25) Confidence Interval = 64 ± 0.41125

Therefore, the 90% confidence interval for the average height of adult women in the city is (63.589 inches, 64.411 inches). We can be 90% confident that the true population mean height falls within this range.

Example 2: Using the T-Distribution

Suppose we want to estimate the average test score of students in a small class. We collect data from 20 students and find that the sample mean score is 75, and the sample standard deviation is 8.

1. Sample Statistic: Sample mean (x̄) = 75
2. Sample Standard Deviation (s): 8
3. Sample Size (n): 20
4. Degrees of Freedom (df): n - 1 = 20 - 1 = 19
5. Critical Value (tα/2, df): Looking up the t-value for α/2 = 0.05 and df = 19 in a t-table, we find a value of approximately 1.729.
6. Standard Error: s / √n = 8 / √20 ≈ 1.789

Now, we can calculate the 90% confidence interval:

Confidence Interval = x̄ ± (tα/2, df * Standard Error) Confidence Interval = 75 ± (1.729 * 1.789) Confidence Interval = 75 ± 3.093

Therefore, the 90% confidence interval for the average test score of students in the class is (71.907, 78.093). We can be 90% confident that the true population mean test score falls within this range.

Interpreting the 90% Confidence Interval

The interpretation of a 90% confidence interval is crucial. It's essential to understand what the interval tells us and what it doesn't.

• Correct Interpretation: We are 90% confident that the true population parameter (e.g., the population mean) lies within the calculated interval. This means that if we were to repeat the sampling process many times and construct a 90% confidence interval for each sample, we would expect 90% of those intervals to contain the true population parameter.
• Incorrect Interpretation: It is incorrect to say that there is a 90% probability that the true population parameter lies within the calculated interval. The population parameter is a fixed value (although unknown), and the interval is what varies from sample to sample. The confidence level refers to the long-run proportion of intervals that would contain the true parameter.

Factors Affecting the Width of the Confidence Interval

Several factors can influence the width of the confidence interval:

• Critical Value: As discussed earlier, a larger critical value (corresponding to a higher confidence level) results in a wider interval.
• Sample Size: Increasing the sample size decreases the standard error, which in turn narrows the confidence interval. Larger samples provide more precise estimates of the population parameter.
• Standard Deviation: A larger standard deviation (or sample standard deviation) indicates greater variability in the data, leading to a wider confidence interval.

Practical Applications of 90% Confidence Intervals

90% confidence intervals are used in various fields to estimate population parameters and make informed decisions. Some examples include:

• Healthcare: Estimating the average effectiveness of a new drug or treatment.
• Marketing: Estimating the average spending habits of customers.
• Finance: Estimating the average return on investment for a particular asset.
• Education: Estimating the average test scores of students in a school district.
• Social Sciences: Estimating the average income level in a particular community.

While 95% confidence intervals are more commonly used, 90% confidence intervals can be appropriate when:

• Lower Confidence is Acceptable: When the consequences of being wrong are not severe, a lower confidence level may be acceptable.
• Need for a Narrower Interval: When a narrower interval is desired, a lower confidence level will result in a smaller margin of error.
• Exploratory Research: In exploratory research, where the goal is to generate hypotheses rather than confirm them, a 90% confidence interval can be useful.

Potential Pitfalls and Considerations

• Assumptions: Confidence intervals rely on certain assumptions about the data, such as normality and independence. Violating these assumptions can lead to inaccurate results. It's crucial to check these assumptions before constructing a confidence interval.
• Outliers: Outliers can significantly affect the sample mean and standard deviation, leading to a wider or biased confidence interval. It's essential to identify and address outliers appropriately.
• Sample Bias: If the sample is not representative of the population, the resulting confidence interval may not accurately reflect the population parameter.
• Misinterpretation: As mentioned earlier, it's crucial to interpret the confidence interval correctly. Avoid the common mistake of stating that there is a certain probability that the true population parameter lies within the interval.

Alternatives to 90% Confidence Intervals

While 90% confidence intervals are useful in certain situations, other confidence levels are also commonly used. The most common alternatives include:

• 95% Confidence Intervals: These are the most widely used confidence intervals, offering a balance between precision and confidence.
• 99% Confidence Intervals: These provide a higher level of confidence but result in wider intervals. They are appropriate when the consequences of being wrong are severe.
• 80% Confidence Intervals: These offer less confidence but result in narrower intervals. They can be useful in situations where a precise estimate is needed, and the consequences of being wrong are not significant.

The choice of confidence level depends on the specific context and the trade-off between precision and confidence.

Conclusion

Understanding the critical value is fundamental to constructing and interpreting confidence intervals accurately. For a 90% confidence interval, the critical value is approximately 1.645 when using the z-distribution and varies depending on the degrees of freedom when using the t-distribution. By understanding the factors that influence the width of the confidence interval and potential pitfalls, researchers and practitioners can use 90% confidence intervals effectively to estimate population parameters and make informed decisions in various fields. Remember to always interpret the confidence interval correctly and consider the assumptions underlying its calculation to ensure the validity of the results.