Confidence Interval

1.A physical therapist wished to estimate, with 99 percent confidence, the mean maximal strength of a particular muscle in a certain group of individuals. He is willing to assume that strength scores are approximately normally distributed with an unknown population standard deviation.A sample of 22 subjects who participated in the experiment yielded a mean of 87.5, and a sample standard deviation of 10.Using this information, develop a 99% confidence interval for the population mean, and interpret the results.[5 marks]

Sample Size

2.A research group would like to conduct a mental health study in a small town to determine the mean number of people who have had mental health issues in their life.The error in estimating the mean is to be less than 0.05 with a 90% level of confidence.No estimate of the population proportion exists.What is the size of the sample needed for this study?[3 marks]

One Sample Hypothesis Testing

3.A psychologist wishes to see if the IQ of people residing in a city with a University and a College is higher than that of the average IQ for people in Canada.The mean IQ for people in Canada is 106, with a standard deviation of 15.The psychologist conducts a study in the city, and finds that the intelligence quotients of a group of 78 residents are normally distributed, with a mean of 110 and a standard deviation of 17.At the 0.01 significance level, is it reasonable to assume that the town's average IQ is greater than the average IQ in Canada?[5 marks]

4.Suppose we know from a study conducted in 2000 that the average female in Canada ages 31 - 49 has an average cholesterol level of 192 mg/dl with a standard deviation of 11.7 mg/dl.A female that has a cholesterol level above 230 mg/dl is considered to have high cholesterol.In 2000, it was estimated that 27% of women in Canada ages 31 - 49 were considered to have high cholesterol levels.We want to know if the percent of women in Canada ages 31 - 49 who are considered to have high cholesterol levels has changed since 2000.A sample of 2300 women in a Canadian city between the ages of 31 - 49 had blood tests done, and it was found that 684 of them had cholesterol levels in the high range (above 230 mg/dl).

At the 0.01 level of significance, can we say if the mean percent of Canadian women between the ages of 31 - 49 who have high cholesterol levels has changed since 2000?[5 Marks]

Two Sample Hypothesis Testing

5.The Gibbs Baby Food Company wishes to compare the weight gain of infants using its brand versus its competitor's.A sample of 40 babies using the Gibbs products revealed a mean weight gain of 3.5 kg in the first three months after birth.For the Gibbs brand, the standard deviation of the population is 1.05 kg.A sample of 55 babies using the competitor's brand revealed a mean increase in weight of 3.7 kg with a standard deviation of 1.3 kg.Can we conclude that babies using the Gibbs brand gained less weight?Choose a 0.05 level of significance, and interpret the results.[5 Marks]

Chai-Square Goodness of Fit

6.The safety director of Honda took samples at random from the file of minor work-related accidents and classified them according to the time the accident took place. [7 Marks]

Time

Number of Accidents (f₀)

Expected Number of Accidents (f_e)

f₀ - f_e

(f₀ - f_e)²

(f₀ - f_e)² / f_e

8 am up to 9 am

6

9 am up to 10 am

6

10 am up to 11 am

20

11 am up to 12 pm

8

1 pm up to 2 pm

7

2 pm up to 3 pm

8

3 pm up to 4 pm

19

4 pm up to 5 pm

6

Total

Using the Goodness-of-fit test and the 0.01 level of significance, determine whether the accidents are evenly distributed throughout the day, and interpret your results.