Question

a statistician wants to obtain a systematic random sample of size 65 from a population of 7285. what is k? 65 to do so they randomly select a number from 1 to k, getting 103. starting with this person, list the numbers corresponding to all people in the sample. 103, , , ...

Explanation:

Step1: Calculate the sampling interval \( k \)

To find the sampling interval \( k \) for a systematic random sample, we divide the population size \( N \) by the sample size \( n \). So \( k=\frac{N}{n}=\frac{7285}{65} = 112.0769\cdots\)? Wait, no, wait, the problem initially thought \( k = 65 \) but that's wrong. Wait, no, systematic sampling: \( k=\lfloor\frac{N}{n}
floor\) or \( k=\lceil\frac{N}{n}
ceil\)? Wait, actually, the formula for \( k \) in systematic sampling is \( k=\frac{N}{n}\) when \( N \) is divisible by \( n \). Let's check \( 65\times112 = 7280 \), \( 65\times113=7345 \) which is more than 7285. Wait, but the problem says "a statistician wants to obtain a systematic random sample of size 65 from a population of 7285". Wait, maybe the initial \( k \) was miscalculated. Wait, no, let's recalculate \( \frac{7285}{65} \). Let's do 65*112 = 7280, 7285 - 7280 = 5. So actually, the correct \( k \) should be \( \lfloor\frac{7285}{65}
floor = 112 \)? Wait, but the problem has a box with 65, maybe that's a mistake. Wait, no, the question first asks "What is k? 65" but that's incorrect. Wait, no, maybe the problem is that \( k \) is the sampling interval, which is \( \frac{N}{n} \). Wait, 7285 divided by 65: 65112 = 7280, 7285 - 7280 = 5. So maybe the statistician is using \( k = 112 \)? Wait, no, the problem says "they randomly select a number from 1 to k, getting 103". Wait, maybe the initial \( k \) was wrong. Wait, let's re-express: in systematic sampling, the sampling interval \( k \) is calculated as \( k=\frac{N}{n} \) (if \( N \) is divisible by \( n \)) or we take the floor or ceiling. Let's compute \( \frac{7285}{65} = 112.0769\), so we take \( k = 112 \) (since 65112 = 7280, and we have 5 extra, but maybe we adjust). But the problem's first box has 65, which is wrong. Wait, maybe the problem has a typo, but let's proceed with the correct \( k \). Wait, no, maybe the user made a mistake, but according to the problem, after selecting 103, the next number is 103 + k. Wait, if \( k \) is 112, then next is 103 + 112 = 215, then 215 + 112 = 327, etc. But let's check the population size: 103 + (65 - 1)112 = 103 + 64112 = 103 + 7168 = 7271, which is less than 7285. Then the last one would be 7271 + 112 = 7383, which is more than 7285, so maybe we take \( k = 112 \) and adjust the last few. But the problem's first box says 65, which is incorrect. Wait, maybe the problem meant \( k = \frac{7285}{65} = 112.07 \), but rounded down to 112. Alternatively, maybe the initial \( k = 65 \) is wrong, and the correct \( k \) is 112. Let's proceed with \( k = 112 \).

Step2: Find the next sample number

Starting with 103, the next number is \( 103 + k \). If \( k = 112 \), then \( 103 + 112 = 215 \). Then the next is \( 215 + 112 = 327 \), then \( 327 + 112 = 439 \), etc.

Wait, but let's check the population size. The population is 7285, sample size 65. So the sampling interval \( k \) should be \( \lfloor\frac{7285}{65}
floor = 112 \) (since 65*112 = 7280, and we have 5 extra units, so we can include those 5 in the sample by adjusting the starting point, but the problem says they selected 103, which is between 1 and k, so k must be at least 103. So \( k = 112 \) is correct because 112 >= 103.

So the first few sample numbers are 103, 103 + 112 = 215, 215 + 112 = 327, 327 + 112 = 439, etc.

Answer:

103, 215, 327, 439, ... (The next number after 103 is 215, then 327, then 439, and so on, adding 112 each time until we reach the 65th term, ensuring it does not exceed 7285.)