# Normal Distribution and Linear Regression

Please see the attached file for the fully formatted problems.

Question 1

During 2002 the number of beds required per day at St Hallam's hospital was normally distributed with a mean of 1800 and a standard deviation of 190. During the first 50 days of 2003 the average daily requirement for beds was 1830. A senior hospital manager claims that this gives evidence that the requirements for beds has changed since 2001. Do you agree?

Question 2

The following data were obtained in an experiment to estimate a possible relation between the number of showings in one week of a TV commercial in typical sales territories and the sales (in thousands of units) in that territory of the advertised article.

Territory 1 2 3 4 5 6 7 8 9 10

Showings 3 1 4 0 2 4 0 3 1 2

Sales 2.6 1.2 3 1 2 3.6 0.5 3.2 1.8 2.5

(i) Show that the data is significantly correlated.

(ii) Construct a scatter pot of the data.

(iii) Obtain a linear regression model for sales against number of showings per week.

(iv) What will be the estimated sales if there are 7 showings in a week? And for 20 showings?

#### Solution Preview

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Question 1

During 2002 the number of beds required per day at St Hallam's hospital was normally distributed with a mean of 1800 and a standard deviation of 190. During the first 50 days of 2003 the average daily requirement for beds was 1830. A senior hospital manager claims that this gives evidence that the requirements for beds has changed since 2001. Do you agree?

In order to answer this question, we need to perform a two tailed z-test since the data is assumed to be normally distributed.

Thus, the first step is to calculate the z-test statistic. However, we must set up an hypothesis. Therefore,

Null Hypothesis, Ho: The requirements for beds have not changed since 2001 (The means are equal)

Alternative Hypothesis, Ha: The requirements for the beds have changed since 2001 ...

#### Solution Summary

Problems involving linear regression and normal distribution are solved. The solution is detailed, well presented and includes a graph.