Determining the confidence interval for the average weight of Allen’s hummingbirds requires gathering weight measurements for a sample of the hummingbirds, calculating the sample mean and standard deviation, and then using those values to calculate the 80% confidence interval. The confidence interval provides a range of likely values for the true average weight, with 80% confidence that the true value lies within that range.

In this article, we will walk through an example where weight measurements were gathered for a sample of 50 Allen’s hummingbirds. We will calculate the sample mean and standard deviation, determine the critical value for an 80% confidence interval, and then calculate the confidence interval. This will demonstrate the steps involved in determining an 80% confidence interval and provide a realistic estimate for the average weight.

## Gathering the Sample Data

The first step in determining the confidence interval is to gather weight measurements for a sample of Allen’s hummingbirds. For this example, we gathered data for a random sample of 50 adult Allen’s hummingbirds. The individual weight measurements are shown in the table below.

2.8 | 3.1 | 2.6 | 3.0 | 2.9 | 2.7 | 3.5 | 3.6 | 2.4 | 3.2 |

3.3 | 2.5 | 3.4 | 3.7 | 2.9 | 3.1 | 2.7 | 3.2 | 2.8 | 3.3 |

3.0 | 2.6 | 2.4 | 2.9 | 3.1 | 3.0 | 2.5 | 2.6 | 3.4 | 2.7 |

3.5 | 2.5 | 2.8 | 3.0 | 3.3 | 3.1 | 3.0 | 2.9 | 2.7 | 2.9 |

2.6 | 3.0 | 3.4 | 2.8 | 3.3 | 3.0 | 2.6 | 3.1 | 2.8 | 2.5 |

This sample provides a good representation of the weight distribution in the overall population of adult Allen’s hummingbirds.

## Calculating the Sample Mean

The next step is to calculate the sample mean. This provides an estimate of the true population mean. The sample mean is calculated by summing all the observations and dividing by the sample size.

For our sample of 50 Allen’s hummingbirds, the sum of all the weights is 147.9 grams. Dividing this by the sample size of 50 gives:

Sample Mean = 147.9 g / 50 = 2.958 g

Therefore, the sample mean weight of the hummingbirds is 2.958 grams. This provides our best estimate of the true population mean weight.

## Calculating the Sample Standard Deviation

In addition to the sample mean, we need the sample standard deviation to calculate the confidence interval. The standard deviation measures the amount of variation in the sample. A larger standard deviation indicates the values are more spread out from the mean.

The calculation for the sample standard deviation is:

s = sqrt[(sum of squared deviations from mean) / (n – 1)]

For our sample, the sum of squared deviations from the mean is 17.9025. With a sample size of 50, plugging this into the formula gives:

s = sqrt(17.9025 / 49) = 0.3654 g

The sample standard deviation is 0.3654 g. This value will be used along with the sample mean to determine the confidence interval.

## Determining the Critical Value

To calculate a confidence interval, we need to determine the critical value from the t-distribution. The critical value depends on the confidence level and the sample size.

For an 80% confidence interval with a sample size of 50, the critical value is 0.69 (from a t-distribution table).

This critical value will be used as the multiplier when calculating the margin of error to determine the confidence interval.

## Calculating the Margin of Error

The margin of error depends on the critical value and the standard deviation. It is calculated as:

Margin of error = Critical value * Standard deviation / sqrt(Sample size)

Plugging in the values from our example:

Margin of error = 0.69 * 0.3654 g / sqrt(50)

Margin of error = 0.085 g

This margin of error reflects the amount of uncertainty in either direction from the sample mean.

## Determining the Confidence Interval

Now we can calculate the 80% confidence interval using the sample mean and the margin of error:

80% Confidence Interval = (Sample mean – Margin of error, Sample mean + Margin of error)

For our example:

80% CI = (2.958 g – 0.085 g, 2.958 g + 0.085 g)

80% CI = (2.873 g, 3.043 g)

Therefore, we can say with 80% confidence that the true average weight of adult Allen’s hummingbirds lies between 2.873 g and 3.043 g.

## Interpreting the Confidence Interval

It is important to properly interpret what a confidence interval means. The 80% confidence level means if we repeated this process of gathering sample weights and calculating the interval numerous times, about 80% of the calculated intervals would contain the true average weight.

This does not mean there is an 80% chance the true average weight lies within the interval. It also does not mean we are 80% certain the true value is within the interval. The true value either is within the interval or it is not.

The interval provides an estimated range for the unknown true value. The wider the interval, the less precise the estimate.

## Conclusion

In this article, we walked through the steps to determine an 80% confidence interval for the average weight of Allen’s hummingbirds. Based on a sample of 50 hummingbirds, we calculated a confidence interval of (2.873 g, 3.043 g). This indicates our best estimate is that the true population average weight lies within that range.

The key steps were:

- Gather sample data
- Calculate sample mean = 2.958 g
- Calculate sample standard deviation = 0.3654 g
- Determine critical value = 0.69
- Calculate margin of error = 0.085 g
- Calculate 80% CI = (2.873 g, 3.043 g)

Properly interpreting a confidence interval and understanding the meaning of confidence levels is important for statistical analysis. Though estimates vary from sample to sample, the confidence interval can provide a reasonable range for unknown population values.