4 only
To determine the values of \(x\), given the set of numbers \(59, 53, 51, 43, 36, (8x + 1), (x^2 + 1), 13, 12, 9, 7, 3\) with a median of 25, follow these steps:
- First, note that there are 12 numbers in total. As the median is the middle value(s), for 12 numbers, the median will be the average of the 6th and 7th numbers when they are in descending order.
- Arrange the non-variable numbers in descending order: \(59, 53, 51, 43, 36, 13, 12, 9, 7, 3\).
- Insert the expressions involving \(x\): \(8x + 1\) and \(x^2 + 1\) into the correct positions in the list. We know these fall around the median, so they must be the 6th and 7th, or adjacent.
- Given the median is 25, it follows that: \(\frac{(8x + 1) + (x^2 + 1)}{2} = 25\).
- Solving this equation:
- This is a quadratic equation: \(x^2 + 8x - 48 = 0\). Solve it using factorization or the quadratic formula:
- Try factorization:
- Evaluate each solution:
- If \(x = 4\), then \(8x + 1 = 33\) and \(x^2 + 1 = 17\). Insert these numbers into our sequence: \(59, 53, 51, 43, 36, 33, 17, 13, 12, 9, 7, 3\). The median is indeed 25 (average of 17 and 33).
- If \(x = -12\), then \(8x + 1 = -95\) and \(x^2 + 1 = 145\). Insert these numbers: \(145, 59, 53, 51, 43, 36, 13, 12, 9, 7, 3, -95\). Here, the median is not 25, so \(x = -12\) is invalid.
- Only \(x = 4\) ensures the median is 25, confirming the correct answer is \(4 \text{ only}\).
Important Questions from Statistics
Methods of presentation of data are:
Consider 35 i.i.d. observations X1, X2, ...., X15 and Y1, Y2, ..., Y20. Let R be the Wilcoxon's rank sum statistic based on the ranks of the X' s in the combined sample. Then the expected value of R is
Let I, J > 5. Consider two-way ANOVA, where the observations satisfy the linear model
yij = α + βi + γj + εij, 1 ≤ i ≤ I, 1 ≤ j ≤ J
E(εij) = 0, Var(εij) = σ2, \(\sum_{i=1}^{!} β_i=\sum_{j=1}^j γ_j=0 .\)
In this set - up
Let X1,X2,...,Xn be a random sample from exponential distribution with mean θ. Which of the following is NOT a sufficient statistic for θ?
A newly developed algorithm for random number generation needs to be tested. The first step is to check whether the sequence of numbers generated can be considered a random sample from the uniform distribution on the interval (0, 1). Which of the following is an appropriate nonparametric test?
