Suppose that you find out that 20% of all babies say their first word by the time they are 13 months old, and that 90% of all babies say their first word by the time they are 22 months old. Assume that the age at which a baby says its first word is normally distributed. Find the mean age at which a baby says its first word, and the standard deviation.
I don't know how to standardize this without either one (the mean or standard deviation) if that is even what I have to do?
Given the mean and atsndard deviation, you can find the score for an area under the curve, but working backward, given two percentiles and scores, you can work that in reverse.
Reading off the numbers from a z-table (I hope you don't have to work it out from the equation!) I see that 90% equates to 1.28 SD away from the mean and 20% equates to 0.84 SD from the mean. (We don't have to worry about sign, since the curve is symmetric.)
So 9 months, being the difference between 13 and 22, constitutes a total of 2.32 SD. Does that give you enough to go on?
I'm not sure where you get the 2.32 SD from? And would I use the percentages or months in the calculation with what mean?
90% is 22 months, is 1.28 SD from the mean.
20% is 13 months, is .84 SD in the opposite direction from the mean.
Thus, a difference of (22 - 13) months is equivalent to to (1.28 - (-.84)) SD.
That equals 2.12. So, I use the z-score from the 20% and 90% and the 2.12 SD, but what do I use for the two means or top numbers when standardizing?
Sorry about my mental arithmetic error. :-)
So 9 months is 2.12 SD. so 1SD is 4.25 months.
1.28 SD is then 5.44 months from the mean, so the mean is 22 - 5.44.
Check: working it the other way .84 is 3.57 months, so the mean is 13 + 3.57.
Do those agree? Yes, our mean is 16.57, with an SD of 4.25.
To find the mean and standard deviation, you'll need to use the information given about the percentage of babies who say their first word by a certain age. However, you don't need to standardize anything in this case.
Let's start by considering the mean age at which a baby says its first word. We know that 20% of babies say their first word by 13 months, and 90% say their first word by 22 months. This means that the bulk of the data falls within this range.
To estimate the mean, we can assume that the distribution is symmetric and centered around the middle of this range, which is 17.5 months ((13 + 22) / 2). This is a reasonable assumption given that the distribution is stated to be normal.
Next, let's consider the standard deviation. In a normal distribution, roughly 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.
Since the range from 13 months to 22 months encompasses 70% (90% - 20%) of the data, we can assume that this range covers approximately two standard deviations. So, each standard deviation covers approximately 35% (70% / 2) of the data.
To find the standard deviation, we need to determine the age at which one standard deviation occurs. We can calculate this by dividing the range (9 months) by the percentage coverage (35%) of each standard deviation.
Calculating: 9 months / 35% = 25.71 months.
Therefore, the mean age at which a baby says its first word is estimated to be 17.5 months, and the standard deviation is approximately 25.71 months.