Assume that 60% of community college students are female.
1) if we pick 100 students at random justify why we can or cannot model this using a normal model.
2) find the probability that between 50 and 70 of the 100 students selected are female.
3) fidn Q1 and Q3 for the middle 50% for the percent of female students chosen out of 100.
1) To determine whether we can model this using a normal model, we need to check if the conditions for applying the normal distribution are met.
The conditions for using a normal model are:
- The population distribution is approximately normal or the sample size is large enough (around 30 or greater).
- The observations in the sample are independent.
In this case, we are assuming that we randomly pick 100 students from a larger population. Since it is not mentioned whether the population distribution is normal, we need to assess the sample size.
With a sample size of 100, if the underlying population distribution is not extremely skewed, we can assume that the distribution of the sample proportions will be approximately normal due to the Central Limit Theorem. Thus, we can model the proportion of female students using a normal distribution.
2) To find the probability that between 50 and 70 of the 100 students selected are female, we need to determine the probability of obtaining a sample proportion within this range.
To calculate this, we need to find the mean and standard deviation of the sample proportion. The mean (μ) of the sample proportion can be estimated as the assumed proportion of females in the population, which is 60% (or 0.60). The standard deviation (σ) of the sample proportion can be calculated using the formula:
σ = sqrt[(p * (1 - p)) / n]
Where p is the assumed proportion, and n is the sample size.
The probability can then be calculated by finding the z-scores for 50 and 70 and using them to calculate the area under the normal curve between these two values.
3) To find Q1 and Q3 for the middle 50% of the percent of female students chosen out of 100, we first need to calculate the lower quartile (Q1) and the upper quartile (Q3).
Q1 represents the value below which lies the first quartile of the data, containing 25% of the observations. Q3 represents the value below which lies the third quartile of the data, containing 75% of the observations.
To calculate Q1 and Q3, we need to find the z-scores corresponding to the 25th and 75th percentiles. We can use the z-table or a calculator to find these values. Once we have the z-scores, we can use the formula:
x = μ + (z * σ)
Where x is the desired data value, μ is the mean, z is the z-score, and σ is the standard deviation. Finally, the values obtained for Q1 and Q3 will represent the boundaries of the middle 50% of the distribution of the percent of female students chosen out of 100.