May 01, 2023

### 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.

### 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 .

1. 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., .
2. 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;
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