The 2020 statistics show that 50% of women age 20 and younger have taken a pregnancy test and 30% of those who have taken a pregnancy test are or have been pregnant. Over the counter pregnancy tests give a false positive 5% of the time and a false negative 13% of the time.
1). Assuming that none of the women who did not take a pregnancy test were pregnant, what percentage of women age 20 and under are or have been pregnant?
2) Given that a woman took a pregnancy test, what is the probability that she had a false negative test?
3) For a woman who received a negative pregnancy test, what is the probability she actually was pregnant?
4) For a woman who received a positive pregnancy test, what is the probability that she was not pregnant?
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1) To determine the percentage of women age 20 and younger who are or have been pregnant, we need to consider two groups: those who have taken a pregnancy test and those who have not.
a) Women who have taken a pregnancy test: According to the statistics, 50% of women age 20 and younger have taken a pregnancy test. Of these, 30% are or have been pregnant. Therefore, the proportion of women who have taken a pregnancy test and are or have been pregnant is 30% of 50%, which is 0.3 * 0.5 = 0.15 or 15%.
b) Women who have not taken a pregnancy test: Assuming that none of the women who did not take a pregnancy test were pregnant, the proportion of women who have not taken a test and are or have been pregnant is 0%.
To find the overall percentage of women age 20 and younger who are or have been pregnant, we add the proportions from both groups: 15% (from those who took a pregnancy test) + 0% (from those who did not take a test) = 15%.
Therefore, the percentage of women age 20 and under who are or have been pregnant is 15%.
2) Given that a woman took a pregnancy test, we want to find the probability that she had a false negative result.
Let's assume 100 women took the pregnancy test. From the statistics, we know that 13% of the tests give a false negative.
Out of 100 women, 30% are or have been pregnant (as calculated in question 1). So, 30 women are actually pregnant.
Now, let's calculate the number of women who had a positive pregnancy test within the group of pregnant women. From the statistics, we know that the test gives a false negative 13% of the time, meaning it correctly detects pregnancy 100% - 13% = 87% of the time. So, out of the 30 pregnant women, 87% will test positive, which is 0.87 * 30 = 26.1. Rounding to the nearest whole number, we have 26 pregnant women who had a positive test.
Therefore, out of the 100 women who took the pregnancy test, 26 had a true positive and are actually pregnant. The remaining 100 - 26 = 74 women had a false negative test.
Hence, the probability that a woman who took the pregnancy test had a false negative is 74 out of 100, or 74%.
3) For a woman who received a negative pregnancy test, we want to find the probability that she actually was pregnant.
Following from question 2, out of the 100 women who took the pregnancy test, 74 had a false negative (as calculated in question 2). Therefore, out of the 100 women, 74 are actually pregnant.
Now, considering only the women who received a negative test, we need to find the proportion of those who are truly pregnant. Since the test gave a false negative to 74 women, the remaining 100 - 74 = 26 women received a true negative.
Out of these 26 women, 0 are actually pregnant (as they received a true negative).
Thus, for a woman who received a negative pregnancy test, the probability that she actually was pregnant is 0 out of 26, or 0%.
4) For a woman who received a positive pregnancy test, we want to find the probability that she was not pregnant.
From the statistics, we know that the over-the-counter pregnancy test gives a false positive 5% of the time.
If we assume 100 women took the pregnancy test, and 30% of them are actually pregnant (as calculated in question 1), we have 30 women who are truly pregnant.
From the remaining women, who are not pregnant, we calculate the number of women who received a false positive test. From the statistics, we know that the test gives a false positive 5% of the time. So, out of the 70 non-pregnant women, 5% will have a false positive, which is 0.05 * 70 = 3.5. Rounding to the nearest whole number, we have 3 non-pregnant women who had a false positive test.
Therefore, out of the 100 women who took the pregnancy test, 3 had a false positive and are not pregnant.
Hence, the probability that a woman who received a positive pregnancy test was not actually pregnant is 3 out of 100, or 3%.