Question

Suppose the mean number of days to germination of a variety of seed is 22, with standard deviation 2.3 days. find the probability that the mean germination time of a sample of 160 seeds will be within 0.5 days of the population mean.

Answer 1

To find the probability that the mean germination time of a sample of 160 seeds will be within 0.5 days of the population mean, we can use the Central Limit Theorem. The probability is 1.9954.

Step-by-step explanation:

To find the probability that the mean germination time of a sample of 160 seeds will be within 0.5 days of the population mean, we can use the Central Limit Theorem. According to the theorem, as the sample size increases, the distribution of the sample mean approaches a normal distribution. In this case, the population mean is 22 days and the standard deviation is 2.3 days.

We can calculate the standard deviation of the sample mean by dividing the population standard deviation by the square root of the sample size. So, the standard deviation of the sample mean is 2.3 / √160 = 0.182.

Next, we need to find the z-scores for the lower and upper limits of the desired range. The z-score for 0.5 days below the population mean can be calculated as (mean - lower limit) / standard deviation = (22 - 21.5) / 0.182 = 2.75.

Similarly, the z-score for 0.5 days above the population mean is (upper limit - mean) / standard deviation = (22.5 - 22) / 0.182 = 2.75.

Using a standard normal distribution table, the probability associated with a z-score of 2.75 is approximately 0.9977.

Since the distribution is symmetric, the probability of the sample mean being within 0.5 days of the population mean is twice this value, i.e., 2 x 0.9977 = 1.9954, or approximately 1.9954.

Therefore, the probability that the mean germination time of a sample of 160 seeds will be within 0.5 days of the population mean is 1.9954.

Answer 2

In Note 6.5 "Example 1" in Section 6.1 "The Mean and Standard Deviation of the Sample Mean" we constructed the probability distribution of the sample mean for samples of size two drawn from the population of four rowers. The probability distribution is:

x-152154156158160162164P(x-)116216316416316216116

Figure 6.1 "Distribution of a Population and a Sample Mean" shows a side-by-side comparison of a histogram for the original population and a histogram for this distribution. Whereas the distribution of the population is uniform, the sampling distribution of the mean has a shape approaching the shape of the familiar bell curve. This phenomenon of the sampling distribution of the mean taking on a bell shape even though the population distribution is not bell-shaped happens in general. Here is a somewhat more realistic example.Suppose we take samples of size 1, 5, 10, or 20 from a population that consists entirely of the numbers 0 and 1, half the population 0, half 1, so that the population mean is 0.5. The sampling distributions are:

n = 1:

x−−P(x−−)00.510.5

n = 5:

x−−P(x−−)00.030.20.160.40.310.60.310.80.1610.03

n = 10:

x−−P(x−−)00.000.10.010.20.040.30.120.40.210.50.250.60.210.70.120.80.040.90.0110.00

n = 20:

x−−P(x−−)00.000.050.000.100.000.150.000.200.000.250.010.300.040.350.070.400.120.450.160.500.18

x−−P(x−−)0.550.160.600.120.650.070.700.040.750.010.800.000.850.000.900.000.950.0010.00

Histograms illustrating these distributions are shown in Figure 6.2 "Distributions of the Sample