1. True or False. Justify for full credit.
(a) The normal distribution curve is always symmetric to 0. Hint: Where is the mean?
(b) If the variance from a data set is zero, then all the observations in this data set are identical. Hint: What does variance mean?
(c) P(AOR A complement) =1, where A complement is the complement of A. Hint: What does complement mean?
(d) In a hypothesis testing, if the p-value is less than the significance level α, we do not have sufficient evidence to reject the null hypothesis. https://onlinecourses.science.psu.edu/statprogram/node/138
(e) The volume of milk in a jug of milk is 128 oz. The value 128 is from a discrete data set. https://statistics.wikidot.com/ch6
The IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.
100 + or - sd (15) =1 sd =? 2 sds = ? 3sds = ?
2.. What is the probability that a randomly selected person has an IQ between 85 and 115?
3. Find the 90th percentile of the IQ distribution. Hint: Using a normal table, Probability = .4500
Z = 1.645 sd
4. If a random sample of 100 people is selected, what is the standard deviation of the sample mean?
standard deviation = sd/n
5. A random sample of 100 light bulbs has a mean lifetime of 3000 hours. Assume that the population standard deviation of the lifetime is 500 hours. Construct a 95% confidence interval estimate of the mean lifetime. Show all work. Just the answer, without supporting work, will receive no credit
xbar - 1.96 SD/SQRT(100) < mean < xbar +1.96 SD/SQRT(100) where xbar = sample mean
xbar = 3000 hours, SD = 500 hours, n = 100
False since normal distribution is symmetric about the mean which may be zero or not. Only for a standard normal distribution is the mean zero.
True since variance is based on deviation of the values from the mean and the only way for variance to be zero is that deviation from mean for all value to be zero which is only possible when all the value are same.
True as A complement means non-occurrence of A. Thus, either an event would occur or not occur, hence all the possible outcomes would fall within these two categories.
False, as when p value is lesser than level of significance, then the relation is significant and hence null hypothesis is rejected whereas the alternate hypothesis is accepted.
False as the milk volume is a continuous data which can assume value in decimals.
2. It is known that the distribution is normal.
Mean(µ) = 100, Standard Deviation (σ) = 15
Here, X1 = 85
Calculated Z1 = (85-100)/15 = -1
Also, X2 = 115
Calculated Z2 = (115-100)/15 = 1
Hence, requisite probability P(-1<Z<1) = 0.841345 - 0.158655 = 0.682689
3. The corresponding z value with a probability of 0.9 from the table comes out to be 1.282
Let the 90th percentile IQ level be defined as X
Then, 1.282 = (X- 100)/15
As a result, X = 1.282*15 + 100 = 119.2
4. Standard deviation of the sample mean = Standard deviation of population/√Sample size = 15/√100 = 1.5
5. Since there are 100 light bulbs, hence, it is a reasonable assumption that the distribution is normal in line with the Central Limit Theorem as per which there should be minimum 30 observations.
Sample mean = 3000 hours
Sample standard deviation = 500 hours
Sample size = 100
Hence, Standard Error (SE) = 500/√100 = 50 hours
Further, the 95% confidence level z value is 1.96
Hence, lower level of 95% confidence interval = 3000 – 1.96*50 = 2,902 hours
Upper level of 95% confidence interval = 3000 + 1.96*50 = 3,098 hours