course material

1. STAT101 Introductory Statistics
Answer:
Question 1
The people’s weights are normally distributed with a mean . On the other hand, the
standard deviation is . Given that 15 people have a weight of 1200kg (average of 80kg
person), we need to determine the probability;
The z-score is;
The probability is 0.1966.
Question 2
The hypothesized proportion is . The sample size is and the number of favorable cases
(number of customers who prefer the new brand) is . Therefore, the proportion of
customers who prefer the new brand is;
The objective is to investigate whether at least 75% of users prefer the new brand.
The hypotheses are;
The test is equivalent to a right-tailed test, for which a z-test for one population proportion
is used.
The significance level is . Therefore, the critical value is;
Thus, the rejection region is .
The test statistic is given by;
Since the test statistic , the null hypothesis is not rejected. It is concluded that there is not
enough evidence to claim that at least 75% of users prefer the new brand.

2. Question 3
The random sample of the visitors to the exhibition is . The sample mean of the amount
spent is and the sample standard deviation is .
The assumption needed is that the sample should be approximately normally distributed.
The assumption is required because the sample size is not sufficiently large. i.e., .
The population standard deviation is not known. Therefore, the student’s t-distribution is
used.
The sample size is . Thus, the number of the degrees of freedom is;
The significance level is . Therefore, the two-tailed critical value is;
The 95% confidence interval is given by;
The confidence interval is .
It can be stated with 95% confidence that the actual mean amount spent at the exhibition is
between $58.44 and $81.56.
To examine whether the population mean is significantly different from $75, a one-sample
t-test is performed. The hypotheses are;
The significance level is and the number of degrees of freedom is . Therefore, the critical
value is;
The rejection region is
The test statistic is given by;
Since the absolute test statistic , the null hypothesis is not rejected. It is concluded that there
is not enough evidence to claim that the population mean is significantly different from $75.
If the population standard deviation was known to be , we would apply a z-statistic instead
of a t-statistic. The confidence interval and hypothesis test would be as follows;
Confidence Interval
The significance level is . Therefore, the critical value is .
The 95% confidence interval is given by;
The confidence interval is

3. It can be stated with 95% confidence that the actual mean amount spent at the exhibition is
between $59.02 and $80.98.
Hypothesis Test
The hypotheses for the z-test used to test whether the mean is significantly different from
$75 are;
The significance level is . Therefore, the critical value is .
The rejection region is
The test statistic is given by;
Since the absolute test statistic , the null hypothesis is not rejected. It is concluded that there
is not enough evidence to claim that the population mean is significantly different from $75.
The p-value for part (e) above is;
Since the p-value is greater than the null hypothesis is not rejected. It is concluded that
there is not enough evidence to claim that the population mean is significantly different
from $75.
Question 4
The sample size is . The average number of cars passing the drive-through is cars and the
standard deviation is cars.
Yes, we can assume that the sample distribution of the sample mean is normal because the
sample size is sufficiently large. i.e., .
The hypotheses for the t-test used to investigate whether at least 300 cars pass the location
during the day are;
The significance level is and the number of degrees of freedom is . Therefore, the critical
value is;
The rejection region is
The test statistic is given by;
Since the test statistic t, the null hypothesis is not rejected. It is concluded that there is not
enough evidence to claim that the number of cars passing through the location is more than
300.

4. If the significance level is 0.1 (10%), the critical value is for degrees of freedom equal to 30
is;
Since the test statistic t, the null hypothesis is not rejected. It is concluded that there is not
enough evidence to claim that the number of cars passing through the location is more than
300.
The Conclusion Is The Same As In Part (B) Above.
Type I error is committed when a true null hypothesis is rejected. In this case, the error
would be committed if it was concluded that more than 300 cars pass through the location
when the reality is that not more than 300 vehicles pass through the location. On the other
hand, type II error is committed when a false null hypothesis is not rejected. Therefore, in
this case, the type II error would be committed if it was concluded that not more than 300
cars pass through the location even though there is enough evidence to suggest that more
than three hundred cars do pass through the location.