Question

3. If a random sample of 53 students was asked for the number of semester hours they are taking this semester. The sample standard deviation was found to be s = 4.7 semester hours. How many more students should be included in the sample to be 99% sure that the sample mean x is within 1 semester hour of the population mean  for all students at this college? i. A marketer is trying two different sales pitches to sell a carpet cleaning service. For his aggressive sales pitch, 175 people were contacted by phone, and 62 of those people bought the cleaning service. For his passive sales pitch, 154 people were contacted by phone, and 45 of those people bought the cleaning service. Does this indicate that there is any difference in the population proportions of people who will buy the cleaning service depending on which sales pitch is used? Use  = 0.05.

Answer 1

3. If a random sample of 53 students was asked for the number of semester hours they are taking this semester. The sample standard deviation was found to be s = 4.7 semester hours. How many more students should be included in the sample to be 99% sure that the sample mean x is within 1 semester hour of the population mean  for all students at this college?

Let n be the sample size required to be 99% sure that the sample mean is within 1 semester hour of the population mean for all students at this college. This indicates that we want to estimate the mean at a 99% confidence level and with a margin of error of 1 semister hour.

Since the sample size of 53 is already greater than 30, we can use the central limit theorem and say that the sampling distribution of mean is normal distribution

The 99% confidence interval indiates a significance level of

The right tail critical value is

Using the standard normal tables, we get for z=2.58, P(Z<2.58)=0.995

Hence

s=4.7 is the sample stamdard deviation of the number of semester hours

We will estimate the population standard deviation using the sample as

The estimated standard error of mean is

The margin of error needed is

The required sample size is 148. We need to collect 148-53=95 more students

ans: We need to collect 95 more students should be included in the sample to be 99% sure that the sample mean is within 1 semester hour of the population mean for all students at this college

i) Let be the true proportion of people who will buy the cleaning service when  the aggressive sales pitch is used and be the true proportion of people who will buy the cleaning service when the passive sales pitch is used

We want to test if  there is any difference in the population proportions of people who will buy the cleaning service depending on which sales pitch is used. That is, we want to test if . (This would be the alternative hypothesis as an alternative hypothesis always has one of inequalities)

The hypotheses are

We have the following information from the sample

The common proportion of people who bought the service is

the standard error of difference between 2 proportions is

The hypothesized value of difference between the 2 proportions is

The test statistic is

This is a 2 tailed test (The alternative hypothesis has "not equal to"). The right tail critical value for is

Using the standard normal tables, we get for z=.196, P(Z<1.96)=0.975.

The critical values are -1.96,+1.96

We will reject the null hypothesis, is the test statistic lies outside the interval ( -1.96,+1.96)

Here, the test statistic is 1.199 and it lies with in the interval ( -1.96,+1.96). Hence we do not reject the null hypothesis

ans: Fail to reject H0. There is no sufficient evidence to indicate that there is any significant difference in the population proportions of people who will buy the cleaning service depending on which sales pitch is used.