Question 1
a. When a researcher studies a process or a phenomenon, it is impossible to involve the whole population in the analysis, therefore the certain assumptions and limitations are necessary. Supposing, the population has the mean μ and the standard deviation σ, and the quantity N. The sampling distribution is the smaller sample with the quantity n (N > n), taken from the population (part of the population), with μ that is equal to the population mean (μ=μ) and the standard deviation, that is calculated as: σM=σn. The sampling distribution of the mean approximately represents the properties of the population and can be used for population study and testing he statistical hypothesis.
b. Result 1. If the sample distribution with the quantity n is taken, the distribution of the mean approaches to the normal distribution. The greater the number of n is, the more the distribution approaches the normal. Therefore, all the properties of the normal distribution can be used to describe a sample.
Result 2. The mean of the sampling distribution is approximately equal to the mean of the normal distribution. Thus, if the researcher takes several samples, their means are equal and approach the population mean.
c. All the researches in statistics deal with the samples and their characteristics. Therefore, the descriptive statistics are obtained, hypothesis are set and tested for the sample distributions, not the population characteristics. Furthermore, in many cases it is impossible to obtain the population means and variance, for example when social research is performed. The central limit theorem provides a scientific basis for the approximation of the sample distribution characteristics to the normal distribution (population) and allows applying the properties of the normal distribution to the samples, as well as testing hypotheses with the sampling distributions.
d. The mean of the population is μ = 500 ml, the standard deviation is σ = 14 ml. The mean fill for the first range is μ1 = 497 ml and the mean fill for the second range is μ2 = 502 ml.
The probability that the can is filled with the certain value is obtained basing on the z-values. The z-value calculates: z=μ-μσ. Therefore, z1=497-50014=-0.21; z2=502-50014=0.14.
The probabilities are obtained from the z-table. These are the probabilities that the can is filled less than μ volume, and it is noted as P(X< μ).
The probability that the can is filled less than 497 ml is: P(X < 497) = 0.4168.
The probability that the can is filled less than 502 ml is: P(X < 502) = 0.5557.
Thus, the probability that the can is filled between 497 and 502 ml is P (497 < X < 502) = P(X < 502) - P(X < 497) = 0.5557 - 0.4168 = 0.1389.
The probability of the can being filled between 497 and 502 ml is 0.1389, or there is a 13.89 % chance that a can is filled with the volume 497 - 502 ml.
Question 2
a. The mean study time is μ=50 hours, and the standard deviation is σM=8 hours. The sample size, which is the number of students studying in the course is n = 50. Therefore, the margin of error calculates as:
ME = z · σn .
ME = 1.96 · 850= 2.21.
The value of the weekly study hours in the Business School at Island University is 50 ± 2.21 hours.
b. We assume that the sample taken for the research is normal. This assumption is true since the number of students in the study is rather high (n = 50), therefore the distribution of the mean is close to the normal.
For 95% confidence interval we take z-value, z = 1.96.
The confidence interval is constructed basing on the mean and margin of error values: Mean ± ME. Then, the range stands for the interval to which the mean value belongs to.
c. The 95% confidence interval informs us that the students spend between 47.78 and 52.21 hours per week studying in the Business School at Island University. Having constructed the confidence interval, we can be 95% sure that the value of the study hours at this university is within 47.78 and 52.21 hours.
Question 3
a. The mean study time in all universities is μ =46 hours (parameter), and it is the parameter value. The mean time of the weekly study time is the statistic value. To test whether the full-time business students at Island University study longer than full-time business students at other universities, it is necessary to set up hypothesis.
The null hypothesis H0: the weekly study time of business students at Island University is the same as weekly study time of business students at other universities; or the weekly study hours are equal: μ=μ. The μ value is the statistic value.
The alternate hypothesis is opposite to the null. The alternate hypothesis H1: the weekly study time of business students at Island University is longer than the weekly study time of business students at other universities; or the weekly study hours at Island University are greater: μ>μ.
The hypothesis is tested with the calculation of the test statistic. For this case, the t-test is calculated:
t=Statistic-Parameterσ=50-468=0.5
Then, the test statistic is compared to the critical value. The critical value is a table value for alpha level 0.05. The value is tcrit(0.05;49) = 1.6.
The appropriate hypothesis is accepted basing on the comparison of the t and tcrit values. If t < tcrit, then the null hypothesis is accepted, and alternate hypothesis is rejected; if t > tcrit, the alternate hypothesis is accepted, and null hypothesis is rejected.
Since t < tcrit (0.5 < 1.6), then null hypothesis is accepted, and μ=μ.
b. The study time for students from Island University and other universities are normally distributed values. The t-value is chosen for one-sided t-test with alpha level 0.05. One-sided t-test is applied because the mean value of the sample is tested for being greater than the population value. On the contrary, two-sided test is applied when the sample mean value is tested for being smaller and greater than the population mean.
c. The statistical test is a scientific proof that the weekly study time of business students at Island University is the same as weekly study time of business students at other universities. In other words, there is no significant difference between the study hours of business students at Island University and other universities. The test proves that the statement of the Business school at Island University is groundless.
