Question

According to the 2018 American Community Survey, 90% of the 2.2 million households in Minnesota have computers. Consider drawing a random sample of n = 225 households from all households in Minnesota. Let p̂ denote the sample proportion of households that have computers.

1. (4 points) State the mean, standard deviation, and the approximate shape for the sampling distribution of p̂ .
2. (4 points) Based on your answer in question 1, what is the approximate probability that p̂ is greater than 0.88 but less than 0.93?
   
   The following R output might be useful:

   pnorm(0.88) = 0.8106,    pnorm(0.93) = 0.8238
   pnorm(1) = 0.8413,       pnorm(1.5) = 0.9332
   pt(1, df=224) = 0.8408,  pt(1.5, df=224) = 0.9325

3. (2 points) For each of parts (a) and (b), circle the best answer (no explanation needed):
   
   1. If the population proportion of households in Minnesota that have computers were 85% instead of 90%, while everything else remains the same, how would the standard deviation of p̂ change?   Increases / Decreases / Remains unchanged.
      
   2. If there were 3.3 million instead of 2.2 million households in Minnesota, while everything else remains the same, how would the standard deviation of p̂ change?   Increases / Decreases / Remains unchanged.

Answer 1

1.

Shape : Normal (Bell shaped) (symmetric)

Mean : 0.9

SD : 0.02

2.

3.a. INCREASES (standard deviation increases as p decreases)

3.b. Remains unchanged. (standard deviation remains unchanged for large population)