The phone rings, and you're in the shower, eating dinner or watching your favorite TV show. You say hello and a strange voice replies, "Good evening. This is John Tweedledee of Tweedledee, Tweedledum. Can you use some tax-free income?" "Oh no," you groan. "It's another stock broker calling." And the odds are you get off the phone as quickly as you can.

But what does that broker do? Well, if he wants to make his fortune in the investment world and he's got a thick skin, he'll swallow the rejection and make another phone call to sell that stock, bond or tax shelter. If he can't, eventually he'll give up and drop out of the brokerage business.

The problem of rejection is so severe, in fact, that managers worry about it during hiring. Terry Wallace, president of Johnston, Lemon & Co., looks for special qualities when interviewing prospective brokers, who are going to have to make some "cold calls" to build a list of clients. "One of the biggest characteristics you try to find in these individuals' lives is how well they have taken rejection," Wallace says. In the several years it takes for brokers to get off the ground, Wallace says, "the rejection you run into in trying to find pools of money or trying to develop your clients . . . is incredible."

Clearly, those who survive and prosper are imbued with a self-confidence that is unshaken by the chorus of "I'm-not-interesteds." When she began work as a broker 10 years ago, Marilyn Lowen, a vice president at Laidlaw, Ansbacher in Rockville, made 100 cold calls a day. "I was very persistent . . . I'm a hopeless optimist," she says. Selling mostly tax-exempt bonds, mutual funds and tax shelters, the one-time math teacher earned $36 in her first week. Today her annual earnings are in six figures, but the climb was not without pain.

"Get down to net," a former boss told Lowen. That meant learning to talk directly and, if necessary, being able to confront a reluctant customer by saying, as Lowen did one night, "What's the matter, Mr. Jones; don't you like dealing with a woman?" The man at the other end of the line, shocked, replied, "Oh no!" To prove it, he became a client.

Lowen has encountered suspicious wives who wonder why a strange woman is calling their husbands; other brokers have been embarrassed to discover that the person they are calling has died. Timing, of course, is important. Michael S. Roberts, a Prudential-Bache Securities Inc. account executive in Bethesda, never calls during a Redskins game or Monday night football. He's also careful about calls after 9 p.m. "We don't want to irritate people."

Roberts, a former newsman, says that because of customer attrition, few brokers can ever stop prospecting. But brokers don't throw darts at the phone book. Prospective clients include people who have answered ads offering investment information or have bought tax shelters; people who hold top government jobs or live in affluent communities also are good propsects. Some lists are compiled by the brokerage houses themselves, others can be purchased from commercial "listing" firms. One broker recently bought a list of 500 people who have between $50,000 and $100,000 in investments. The cost: 29 cents a name. The company vowed not to resell the list for six months.

Whatever list brokers use, Roberts says, the rule is: Keep going. When someone says, "I'm not interested," the broker should say, "Thank you, goodbye" and move on to the next call. "It's a numbers type of game."

The real bane of a broker's sales effort, Roberts adds, is not the person who is hostile -- Roberts says he has had relatively few of those -- but the person who is eager to pick his brains for all sorts of investment advice, takes a half-hour of his time, and then won't buy.

That's when it's the broker's turn to say, "Sorry, I've got to go." Live by the phone, perish by the phone.

Speaking of prospecting, Ian M. T. McAvity, one of the world's foremost "goldbugs," was in town last week to talk about gold prices -- which are way down -- and their prospects, which are uncertain. With breezy charm, McAvity took his audience at the Washington Marriott Hotel on a nimble journey through the intricacies of gold investing, recommending that portfolios contain 5 percent gold-related investments. "Put two Krugerrands or Maple Leafs under your pillow and I think you're going to sleep a little better at night," McAvity quipped. McAvity, who publishes a newsletter on gold in Toronto, was brought to Washington by Prudential-Bache, which, not coincidentally, is offering $20 units in Gold Co. of America, a partnership between Prudential-Bache and Barrick Resources of Canada. The units will be traded over-the-counter and the sponsors suggest that annual returns to investors and the price of the units both will rise as the price of gold goes up -- if and when it goes up.

"Fairchild has been a dull performer this year," reports Drexel Burnham Lambert Inc. in an assessment of the aerospace firm. So dull, in fact, that the Drexel Burnham analysts have lowered their earnings estimates for Fairchild Industries from $1.60 to $1.55 for 1984; from $1.70 to $1.60 for 1985, and from $2.60 to $2.35 for 1986.

"Fairchild has had the worst of all possible worlds," the analysts declare. "It is out of sync with the rearmament cycle of the nation. The A-10 program, which accounted for over 50 percent of earnings in 1979 and 1980, peaked out in 1980, and deliveries ended this year. The company's commercial aerospace activities . . . peaked out at the same time and have been generating heavy losses since 1980."

Although the analysts have lowered their estimates, they suggest that Fairchild's earnings could hit $4 to $5 a share by 1988 or 1989 as a new Air Force training plane gets into production and as business picks up in the general aviation, industrial and telecommunication sectors. As many as 650 aircraft worth between $1.5 billion and $2 billion could be involved in the Air Force program through 1992, they said.

Fairchild stock, the analysts note, has been selling at around $17 a share, half its high of $34 in early 1981. Since then, they add, "many of the securities we monitor in the aerospace group have doubled or tripled."

Federal Realty Investment Trust of Chevy Chase, now listed on the American Stock Exchange, is moving to the New York Stock Exchange this month. The firm, headed by Steven J. Guttman, buys and rehabilitates real estate properties, especially shopping centers, around the country. Guttman says that since 1980, assets have grown from $40 million to $140 million and annual earnings have climbed from $2 million to $10 million. Moving to the Big Board, Guttman says, will increase the firm's stature in the business and investment communities.

Citizens Bancorp, the holding company for Citizens Bank of Maryland, occasionally needs stock for its dividend reinvestment plan. Such stock is usually bought on the open market. But because the stock is not heavily traded, there is not always a sufficient amount of stock available. So the bank now will permit negotiated stock purchases with private individuals as well as open-market sales. The program could cover as many as 50,000 shares.