The following data represent the age (in weeks) at which babies first crawl based on a survey of 12 mothers.
52, 30, 44, 35, 47, 37, 56, 26, 35, 35, 52, 52 __________________________________________________________________________
Construct a 95% confidence interval for the mean age at which a baby first crawls. Select the correct choice below and fill in any answers if needed. (choose from one of the below.)
a) (____ , ____)
b) A 95% confidence interval cannot be constructed.
95% = mean ± 1.96 SEm
SEm = SD/√n
Find the mean first = sum of scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
I'll let you do the calculations.
If you can't, then you will have to choose b.
To construct a confidence interval for the mean age at which a baby first crawls, you can use the following steps:
1. Calculate the sample mean: Add up all the observed ages and divide by the total number of observations. In this case, the sum of the ages is (52+30+44+35+47+37+56+26+35+35+52+52) = 541. Divide this by 12 (the number of observations) to get a sample mean of 45.083.
2. Calculate the sample standard deviation: Subtract the sample mean from each observation, square the differences, sum them all up, divide by (n-1), and then take the square root. Using the ages provided, the calculations are as follows:
(52-45.083)^2 + (30-45.083)^2 + (44-45.083)^2 + (35-45.083)^2 + (47-45.083)^2 + (37-45.083)^2 + (56-45.083)^2 + (26-45.083)^2 + (35-45.083)^2 + (35-45.083)^2 + (52-45.083)^2 + (52-45.083)^2 = 1016.166
Then, divide 1016.166 by (12-1) = 1016.166/11 = 92.378.
Lastly, take the square root of 92.378 to get a sample standard deviation of approximately 9.614.
3. Determine the critical value: To construct a 95% confidence interval, you need to determine the critical value from a t-distribution with (n-1) degrees of freedom. Since the sample size is 12, the degrees of freedom is 12-1 = 11. From a t-distribution table or using statistical software, you would find the critical value associated with a 95% confidence level and 11 degrees of freedom, which is approximately 2.201.
4. Calculate the margin of error: Multiply the critical value by the sample standard deviation and divide by the square root of the sample size. In this case, the calculations are as follows:
2.201 * (9.614 / sqrt(12)) = 6.231.
5. Construct the confidence interval: Take the sample mean and subtract the margin of error, and then also add the margin of error to the sample mean. In this case, the calculations are as follows:
Lower bound = 45.083 - 6.231 = 38.852.
Upper bound = 45.083 + 6.231 = 51.314.
Therefore, the 95% confidence interval for the mean age at which a baby first crawls is (38.852, 51.314). The correct choice would be a) (38.852, 51.314).
To construct a 95% confidence interval for the mean age at which a baby first crawls, we can use the following formula:
Confidence Interval = Mean ± (Critical Value) * (Standard Deviation / √n)
Where:
- Mean is the average age at which babies first crawl.
- Critical Value is the value corresponding to the desired confidence level (in this case, 95%), which can be obtained from the t-distribution table.
- Standard Deviation is the measure of the variation in the data.
- n is the sample size.
To find the critical value, we need to calculate the degrees of freedom (df) which is given by (n - 1).
Now let's calculate each component of the formula step-by-step:
Step 1: Calculate the Mean
To find the mean, sum up all the data points and divide by the number of data points:
52 + 30 + 44 + 35 + 47 + 37 + 56 + 26 + 35 + 35 + 52 + 52 = 501
Mean = 501 / 12 = 41.75
Step 2: Calculate the Standard Deviation
To calculate the standard deviation, we need to find the variance first. Variance is given by the sum of squared differences between each data point and the mean, divided by the number of data points minus 1 (n-1):
Variance = Σ(x - x̄)² / (n - 1)
= [(52 - 41.75)² + (30 - 41.75)² + ... + (52 - 41.75)²] / 11
= 605.75
Standard Deviation = √Variance
= √605.75 ≈ 24.61
Step 3: Calculate the Critical Value
The critical value depends on the desired confidence level and the degrees of freedom. Since the number of data points is 12, the degrees of freedom is (12 - 1) = 11. Considering a 95% confidence level, the critical value can be found from the t-distribution table or a statistical calculator. For a sample size of 12 and 95% confidence level, the critical value is approximately 2.201.
Step 4: Calculate the Confidence Interval
Using the formula, Confidence Interval = Mean ± (Critical Value) * (Standard Deviation / √n)
Confidence Interval = 41.75 ± (2.201) * (24.61 / √12)
Confidence Interval = 41.75 ± (2.201) * (24.61 / 3.464)
Confidence Interval = 41.75 ± (2.201) * 7.098
Confidence Interval = 41.75 ± 15.61
Finally, the 95% confidence interval for the mean age at which a baby first crawls is (26.14, 57.36).
Therefore, the correct choice is:
a) (26.14, 57.36)