In a random sample of 50 babies 4 to 7 month old the average number of hours they slept per day turned out to be 14.2 hours with a standard deviation of 1.6 hours find and interpret a 90% confidence interval for the mean number of hours slept by babies at this age
To find the 90% confidence interval for the mean number of hours slept by babies at this age, you can use the formula:
Confidence Interval = Sample Mean ± (Critical Value * Standard Error)
Step 1: Find the Critical Value
Since the sample size is greater than 30, we can use the Z-distribution. For a 90% confidence level, the corresponding critical value is approximately 1.645. This value can also be obtained from a standard normal distribution table or using statistical software.
Step 2: Calculate the Standard Error
The standard error is a measure of the variability of the sample mean. It can be calculated using the formula:
Standard Error = Standard Deviation / √(Sample Size)
In this case, the standard deviation is 1.6 hours and the sample size is 50. Plugging these values into the formula:
Standard Error = 1.6 / √(50)
Step 3: Calculate the Confidence Interval
Now that we have the critical value (1.645) and the standard error, we can calculate the confidence interval using the formula mentioned earlier:
Confidence Interval = Sample Mean ± (Critical Value * Standard Error)
First, calculate the lower bound of the confidence interval:
Lower bound = Sample Mean - (Critical Value * Standard Error)
Lower bound = 14.2 - (1.645 * (1.6 / √(50)))
Next, calculate the upper bound of the confidence interval:
Upper bound = Sample Mean + (Critical Value * Standard Error)
Upper bound = 14.2 + (1.645 * (1.6 / √(50)))
Interpreting the Confidence Interval:
The result will be a range of values that estimates where the true population mean lies, with 90% confidence. For example, if the confidence interval is (13.8, 14.6), it means we are 90% confident that the true mean number of hours slept by babies at this age falls between 13.8 and 14.6 hours.
Keep in mind that this is a random sample, and if you were to take multiple random samples of the same size, around 90% of the intervals calculated would contain the true population mean.